Probability Seminar

Probability Seminars 2015

The seminar is jointly organized between Temple, UPenn and Univ Delaware, by Brian Rider (Temple), Robin Pemantle (Penn), Nayantara Bhatnagar (Delaware).

 

Talks are Tuesdays 2:30 - 3:30 pm and are held either in Wachman 617 (Temple) or David Rittenhouse Lab 3C8 (Penn).

You can also check out the snazzier website maintained by R. Pemantle.

Probability Seminars 2016

The seminar is jointly organized between Temple and Penn, by Brian Rider (Temple) and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman 617 (Temple) or David Rittenhouse Lab 4C6 (Penn).

You can also check out the seminar website at Penn.

  • Tuesday January 26, 2016 at 14:30, David Rittenhouse Lab 3C8

    Largest eignevalues in random matrix beta ensembles: structure of the limit

    Vadim Gorin, MIT

  • Tuesday February 2, 2016 at 14:30, David Rittenhouse Lab 3C8

    Bootstrap percolation on the Hamming torus

    Eric Slivken, UC Davis

  • Tuesday February 9, 2016 at 14:30, David Rittenhouse Lab 3C8

    Limit theorems for monotone subsequences in Mallows permutations

    Nayantara Bhatnagar, Univ Delaware

  • Tuesday February 16, 2016 at 14:30, Wachman 617

    Burgers equations with random forcing

    Yuri Bakhtin, NYU

  • Tuesday March 1, 2016 at 14:30, David Rittenhouse Lab 3C8

    Mean field Ising models 

    Sumit Mukherjee, Columbia

  • Tuesday March 15, 2016 at 14:30, Wachman 617

    The chemical distance in critical percolation

    Philippe Sosoe, Harvard

  • Tuesday March 22, 2016 at 14:30, Wachman 617

    The scaling limit of the loop-erased random walk Green's function

    Christian Benes, CUNY

  • Tuesday March 29, 2016 at 14:30, David Rittenhouse Lab 3C8

    Random walks on abelian sandpiles

    John Pike, Cornell

  • Tuesday April 5, 2016 at 14:30, David Rittenhouse Lab 3C8

    Nodal sets of random eigenfunctions of the harmonic oscillator

    Boris Hanin, MIT

  • Tuesday April 12, 2016 at 14:30, David Rittenhouse Lab 3C8

    Stochastic approach to anomalous diffusion in two dimensional, incompressible, periodic, celluar flows

    Zsolt Pajor-Gyulai, Courant

  • Tuesday April 19, 2016 at 14:30, David Rittenhouse Lab 3C8

    Markov chain convergence via regeneration

    Dan Jerison, Cornell

  • Tuesday April 26, 2016 at 14:30, David Rittenhouse Lab 3C8

    The frog model with drift on R

    Josh Rosenberg, Penn

  • Tuesday September 6, 2016 at 15:00, UPenn (DRL 4C6)

    Random planar metrics of Gaussian free fields

    Jian Ding, Chicago

    I will present a few recent results on random planar metrics of two-dimensional discrete Gaussian free fields, including Liouville first passage percolation, the chemical distance for level-set percolation and the electric effective resistance on an associated random network. Besides depicting a fascinating picture for 2D GFF, these metric aspects are closely related to various models of planar random walks.

     

  • Tuesday September 13, 2016 at 15:00, UPenn (DRL 4C6)

    An Introduction to Limit Theorems for Nonconventional Sums

    Yuri Kifer, Hebrew University

    I'll survey some of the series results on limit theorems for nonconventional sums of the form \[ \sum_{n=1}^NF(X_n,X_{2n},...,X_{\ell n}) \] and more general ones, where $\{ X_n\}$ is a sequence of random variables with sufficiently weak dependence.

  • Tuesday September 20, 2016 at 15:00, Temple (Wachman 617)

    Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension

    Wei Wu, NYU

    Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and $\Phi^4$ models for $d \geq 4$. We describe a simple spin model from uniform spanning forests in $\mathbb{Z}^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension $4$, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in $d=4$, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some "continuum escaping probability". Based on joint works with Greg Lawler and Xin Sun.

     

  • Tuesday September 27, 2016 at 15:00, UPenn (DRL 4C6)

    Corners in tree-like tableaux

    Amanda Lohss, Drexel

    Tree–like tableaux are combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree–like tableaux and the total number of corners in symmetric tree–like tableaux. We have proven both conjectures based on a bijection with permutation tableaux and type–B permutation tableaux. In addition, we have shown that the number of diagonal boxes in symmetric tree–like tableaux is asymptotically normal and that the number of occupied corners in a random tree–like tableau is asymptotically Poisson. This extends earlier results of Aval, Boussicault, Nadeau, and Laborde Zubieta, respectively.

     

  • Tuesday October 4, 2016 at 15:00, Temple (Wachman 617)

    Chaining, interpolation, and convexity

    Ramon van Handel, Princeton

    A significant achievement of modern probability theory is the development of sharp connections between the boundedness of random processes and the geometry of the underlying index set. In particular, the generic chaining method of Talagrand provides in principle a sharp understanding of the suprema of Gaussian processes. The multiscale geometric structure that arises in this method is however notoriously difficult to control in any given situation. In this talk, I will exhibit a surprisingly simple but very general geometric construction, inspired by real interpolation of Banach spaces, that is readily amenable to explicit computations and that explains the behavior of Gaussian processes in various interesting situations where classical entropy methods are known to fail.

     

  • Tuesday October 11, 2016 at 15:00, UPenn (DRL 4C6)

    The front location for branching Brownian motion with decay of mass

    Louigi Addario-Berry, McGill

    I will describe joint work with Sarah Penington (Oxford). Consider a standard branching Brownian motion whose particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle $x$, then the mass of particle $x$ decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical mass-m particles. One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM, and martingales are hard to come by. Nonetheless, it is possible to prove that (in a rather weak sense) the front is at distance $\sim c t^{1/3}$ behind the typical BBM front. At a high level, our argument for this may be described as a proof by contradiction combined with fine estimates on the probability Brownian motion stays in a narrow tube of varying width.

     

  • Tuesday October 18, 2016 at 15:00, UPenn (DRL 4C6)

    Random discrete structures: Scaling limits and universality

    Sanchayan Sen, Eindhoven

    One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on n vertices and degree exponent $\tau>3$, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}$. In other words, the degree exponent determines the universality class the random graph belongs to. More generally, recent research has provided strong evidence to believe that several objects constructed on a wide class of random discrete structures including (a) components under critical percolation, (b) the vacant set left by a random walk, and (c) the minimal spanning tree, viewed as metric measure spaces converge, after scaling the graph distance, to some random fractals, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures. Based on joint work with Shankar Bhamidi, Nicolas Broutin, Remco van der Hofstad, and Xuan Wang.

     

  • Tuesday October 25, 2016 at 15:00, UPenn (DRL 4C6)

    Asymptotics of stochastic particle systems via Schur generating functions

    Alexey Bufetov, MIT

    We will discuss a new approach to the analysis of the global behavior of stochastic discrete particle systems. This approach links the asymptotics of these systems with properties of certain observables related to the Schur symmetric functions. As applications of this method, we prove the Law of Large Numbers and the Central Limit Theorem for various models of random lozenge and domino tilings, non-intersecting random walks, and decompositions of tensor products of representations of unitary groups. Based on joint works with V. Gorin and A. Knizel.

     

  • Tuesday November 1, 2016 at 15:00, UPenn (DRL 4C6)

    Markov Chains of Exchangeable Structures

    Henry Towsner, UPenn

    The Aldous-Hoover Theorem characterizes arrays of random variables which are exchangeable - that is, the distribution is invariant under permutations of the indices of the array. We consider the extension to exchangeable Markov chains. In order to give a satisfactory classification, we need an extension of the Adous-Hoover Theorem to "relatively exchangeable" arrays, which are only invariant under some permutations. Different families of permutations lead to different characterization theorems, with the crucial distinction coming from a model theoretic characterization of the way finite arrays can be amalgamated.

     

  • Tuesday November 8, 2016 at 15:00, UPenn (DRL 4C6)

    Local max-cut in smoothed polynomial time

    Sébastien Bubek, Microsoft

    The local max-cut problem asks to find a partition of the vertices in a weighted graph such that the cut weight cannot be improved by moving a single vertex (that is the partition is locally optimal). This comes up naturally, for example, in computing Nash equilibrium for the party affiliation game. It is well-known that the natural local search algorithm for this problem might take exponential time to reach a locally optimal solution. We show that adding a little bit of noise to the weights tames this exponential into a polynomial. In particular we show that local max-cut is in smoothed polynomial time (this improves the recent quasi-polynomial result of Etscheid and Roglin). Joint work with Omer Angel, Yuval Peres, and Fan Wei.

     

  • Tuesday November 15, 2016 at 15:00, Temple (Wachman 617)

    The law of fractional logarithm in the GUE minor process

    Elliot Paquette, Ohio State

    Consider an infinite array of standard complex normal variables which are independent up to Hermitian symmetry. The eigenvalues of the upper-left NxN submatrices, form what is called the GUE minor process. This largest-eigenvalue process is a canonical example of the Airy process which is connected to many other growth processes. We show that if one lets N vary over all natural numbers, then the sequence of largest eigenvalues satisfies a 'law of fractional logarithm,' in analogy with the classical law of iterated logarithm for simple random walk. This GUE minor process is determinantal, and our proof relies on this. However, we reduce the problem to correlation and decorrelation estimates that must be made about the largest eigenvalues of pairs of GUE matrices, which we hope is useful for other similar problems.

    This is joint work with Ofer Zeitouni.

     

  • Tuesday November 29, 2016 at 15:00, Temple (Wachman 617)

    Arm events in invasion percolation

    Jack Hanson, CUNY

    Invasion percolation is a "self-organized critical" distribution on random subgraphs of Z^2, believed to exhibit much of the same behavior as critical percolation models. Self-organization means that this happens spontaneously without tuning some parameter to a critical value. In two dimensions, some aspects of the invasion graph are known to correspond to those in critical models, and some differences are known. We will discuss new results on the probabilities of various "arm events" -- events that connections from the origin to a large distance n are either present or "closed" in the invasion graph. We show that some of these events have probabilities obeying power laws with the same power as in the critical model, while all others differ from the critical model's by a power of n.

     

Probability Seminars 2017

The seminar is jointly organized between Temple and Penn, by Brian Rider (Temple) and Robin Pemantle (Penn).

For a chronological listing, click the year above.

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn).

You can also check out the seminar website at Penn.

  • Tuesday January 24, 2017 at 15:00, UPenn (David Riitenhouse Lab 3C8)

    Abelian squares and their progenies

    Charles Burnette, Drexel University

    A polynomial P ∈ C[z1, . . . , zd] is strongly Dd-stable if P has no zeroes in the closed unit polydisc D d . For such a polynomial define its spectral density function as SP (z) = P(z)P(1/z) −1 . An abelian square is a finite string of the form ww0 where w0 is a rearrangement of w. We examine a polynomial-valued operator whose spectral density function’s Fourier coefficients are all generating functions for combinatorial classes of con- strained finite strings over an alphabet of d characters. These classes generalize the notion of an abelian square, and their associated generating functions are the Fourier coefficients of one, and essentially only one, L2 (T d)-valued operator. Integral representations and asymptotic behavior of the coefficients of these generating functions and a combinatorial meaning to Parseval’s equation are given as consequences.

     

  • Tuesday January 31, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

    How round are the complementary components of planar Brownian motion?

    Nina Holden, MIT

    Consider a Brownian motion $W$ in the complex plane started from $0$ and run for time $1$. Let $A(1), A(2),...$ denote the bounded connected components of $C-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i \in N$. Our main result is that $E[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty$ for any $\theta <1$. We also prove that $\sum_i r(i)^2|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape. Based on joint work with Serban Nacu, Yuval Peres, and Thomas S. Salisbury.

     

  • Tuesday February 7, 2017 at 15:00, Temple (Wachmann Hall 617)

    Stochastic areas and Hopf fibrations

    Fabrice Baudoin, University of Connecticut

    We define and study stochastic areas processes associated with Brownian motions on the complex symmetric spaces ℂℙn and ℂℍn. The characteristic functions of those processes are computed and limit theorems are obtained. For ℂℙn the geometry of the Hopf fibration plays a central role, whereas for ℂℍn it is the anti-de Sitter fibration. This is joint work with Jing Wang (UIUC).

     

  • Tuesday February 14, 2017 at 15:00, Temple (Wachmann Hall 617)

    Intermediate disorder limits for multi-layer random polymers

    Mihai Nica, NYU

    The intermediate disorder regime is a scaling limit for disordered systems where the inverse temperature is critically scaled to zero as the size of the system grows to infinity. For a random polymer given by a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this talk, I consider polymers made up of multiple non-intersecting walkers and consider the same type of limit. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. This result proves a conjecture about the KPZ line ensemble. Part of this talk is based on joint work with I. Corwin.

     

  • Tuesday February 21, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

    Large deviation and counting problems in sparse settings

    Shirshendu Ganguly, Berkeley

    The upper tail problem in the Erdös-Rényi random graph $G \sim G(n,p)$, where every edge is included independently with probability $p$, is to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1 + d$. The arithmetic analog considers the count of arithmetic progressions in a random subset of $Z/nZ$, where every element is included independently with probability $p$. In this talk, I will describe some recent results regarding the solution of the upper tail problem in the sparse setting i.e. where $p$ decays to zero, as $n$ grows to infinity. The solution relies on non-linear large deviation principles developed by Chatterjee and Dembo and more recently by Eldan and solutions to various extremal problems in additive combinatorics.

     

  • Tuesday February 28, 2017 at 15:00, Temple (Wachmann Hall 617)

    Bounds on the maximum of the density for certain linear images of independent random variables

    James Melbourne, University of Delaware

    We will present a generalization of a theorem of Rogozin that identifies uniform distributions as extremizers of a class of inequalities, and show how the result can reduce specific random variables questions to geometric ones. In particular, by extending "cube slicing" results of K. Ball, we achieve a unification and sharpening of recent bounds on densities achieved as projections of product measures due to Rudelson and Vershynin, and the bounds on sums of independent random variable due to Bobkov and Chistyakov. Time permitting we will also discuss connections with generalizations of the entropy power inequality.

     

  • Tuesday March 21, 2017 at 15:00, Temple (Wachmann Hall 617)

    Local extrema of random matrices and the Riemann zeta function 

    Paul Bourgade, NYU

    Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to conjecture the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on a joint works with Arguin, Belius, Radziwill and Soundararajan.

     

  • Tuesday March 28, 2017 at 15:00, Temple (Wachman Hall 617)

    Majority dynamics on the infinite 3-regular tree

    Arnab Sen, University of Minnesota

    The majority dynamics on the infinite 3-regular tree can be described as follows. Each vertex of the tree has an i.i.d. Poisson clock attached to it, and when the clock of a vertex rings, the vertex looks at the spins of its three neighbors and flips its spin, if necessary, to come into agreement with majority of its neighbors. The initial spins of the vertices are taken to be i.i.d. Bernoulli random variables with parameter p. In this talk, we will discuss a couple of new results regarding this model. In particular, we will show that the limiting proportion of ‘plus’ spins in the tree is continuous with respect to the initial bias p. A key tool in our argument is the mass transport principle. The talk is based on an ongoing work with M. Damron.

     

  • Tuesday April 4, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

    Galton-Watson fixed points, tree automata, and interpretations

    Tobias Johnson, NYU

    onsider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

     

  • Tuesday April 11, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

    Biased random permutations are predictable (proof of an entropy conjecture of Leighton and Moitra)

    Patrick Devlin, Rutgers

    Suppose F is a random (not necessarily uniform) permutation of {1, 2, ... , n} such that |Prob(F(i) < F(j)) -1/2| > epsilon for all i,j. We show that under this assumption, the entropy of F is at most (1-delta)log(n!), for some fixed delta depending only on epsilon [proving a conjecture of Leighton and Moitra]. In other words, if (for every distinct i,j) our random permutation either noticeably prefers F(i) < F(j) or prefers F(i) > F(j), then the distribution inherently carries significantly less uncertainty (or information) than the uniform distribution.

    Our proof relies on a version of the regularity lemma, a combinatorial bookkeeping gadget, and a few basic probabilistic ideas. The talk should be accessible for any background, and we will gently recall any relevant notions (e.g., entropy) as needed. Those unhappy with the talk are welcome to form an unruly mob to depose the speaker, and pitchforks and torches will be available for purchase.

    This is from a recent paper joint with Huseyin Acan and Jeff Kahn.

     

  • Tuesday April 25, 2017 at 15:00, Temple (Wachman Hall 617)

    An introduction to p-adic electrostatics

    Christopher Sinclair, University of Oregon

    We consider the distribution of N p-adic particles with interaction energy proportional to the log of the p-adic distance between two particles. When the particles are constrained to the ring of integers of a local field, the distribution of particles is proportional to a power of the p-adic absolute value of the Vandermonde determinant. This leads to a first question: What is the normalization constant necessary to make this a probability measure? This sounds like a triviality, but this normalization constant as a function of extrinsic variables (like number of particles, or temperature) holds much information about the statistics of the particles. Viewed another way, this normalization constant is a p-adic analog of the now famous Selberg integral. While a closed form for this seems out of reach, I will derive a remarkable identity that may hold the key to unlocking more nuanced information about p-adic electrostatics. Along the way we’ll find an identity for the generating function of probabilities that a degree N polynomial with p-adic integer coefficients split completely. Joint work with Jeff Vaaler.

     

  • Tuesday May 2, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)

    Percolation in Weighted Random Connection Model

    Milan Bradonjic, Bell Labs

    When modeling the spread of infectious diseases, it is important to incorporate risk behavior of individuals in a considered population. Not only risk behavior, but also the network structure created by the relationships among these individuals as well as the dynamical rules that convey the spread of the disease are the key elements in predicting and better understanding the spread. We propose the weighted random connection model, where each individual of the population is characterized by two parameters: its position and risk behavior. A goal is to model the effect that the probability of transmissions among individuals increases in the individual riskfactors, and decays in their Euclidean distance. Moreover, the model incorporates a combined risk behavior function for every pair of theindividuals, through which the spread can be directly modeled or controlled. The main results are conditions for the almost sure existence of an infinite cluster in the weighted random connection model. We use results on the random connection model and sitepercolation in Z^2.

     

  • Tuesday September 5, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

    Large deviations for first passage percolation

    Allan Sly, Princeton

    We establish a large deviation rate function for the upper tail of first passage percolation answering a question of Kesten who established the lower tail in 1986. Moreover, conditional on the large deviation event, we show that the minimal cost path is delocalized, that is it moves linearly far from the straight line path. Joint work with Riddhipratim Basu (Stanford/ICTS) and Shirshendu Ganguly (UC Berkeley).

  • Tuesday September 12, 2017 at 15:00, Temple (Wachman Hall 617)

    Stability of phases and interacting particle systems

    Nick Crawford, Technion

    I will discuss recent work with W. de Roeck on the following natural question: Given an interacting particle system are the stationary measures of the dynamics stable to small (extensive) perturbations? In general, there is no reason to believe this is so and one must restrict the class of models under consideration in one way or another. As such, I will focus in this talk on the simplest setting for which one might hope to have a rigorous result: attractive Markov dynamics (without conservation laws) relaxing at an exponential rate to its unique stationary measure in infinite volume. In this case we answer the question affirmatively.

    As a consequence we show that ferromagnetic Ising Glauber dynamics is stable to small, non-equilibrium perturbations in the entire uniqueness phase of the inverse temperature/external field plane. It is worth highlighting that this application requires new results on the (exponential) rate of relaxation for Glauber dynamics defined with non-zero external field.

  • Tuesday September 19, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

    Invasion percolation on Galton-Watson trees 

    Marcus Michelin, UPenn

    Given an infinite rooted tree, how might one sample, nearly uniformly, from the set of paths from the root to infinity? A number of methods have been studied including homesick random walks, or determining the growth rate of the number of self-avoiding paths. Another approach is to use percolation. The model of invasion percolation almost surely induces a measure on such paths in Galton-Watson trees, and we prove that this measure is absolutely continuous with respect to the limit uniform measure as well as other properties of invasion percolation. This work in progress is joint with Robin Pemantle and Josh Rosenberg.

  • Tuesday September 26, 2017 at 16:00, UPenn (David Rittenhouse Lab 4C8)

    Cutoff for random to random 

    Evita Nestoridi, Princeton

    Random to random is a card shuffling model that was created to study strong stationary times. Although the mixing time of random to random has been known to be of order n log n since 2002, cutoff had been an open question for many years, and a strong stationary time giving the correct order for the mixing time is still not known. In joint work with Megan Bernstein, we use the eigenvalues of the random to random card shuffling to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4} n log n$, answering a conjecture of Diaconis.

  • Tuesday October 3, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

    Lattice path enumeration, multivariate singularity analysis, and probability theory

    Stephen Melczar, UPenn

    The problem of enumerating lattice paths with a fixed set of allowable steps and restricted endpoint has a long history dating back at least to the 19th century. For several reasons, much research on this topic over the last decade has focused on two dimensional lattice walks restricted to the first quadrant, whose allowable steps are "small" (that is, each step has coordinates +/- 1, or 0). In this talk we relax some of these conditions and discuss recent work on walks in higher dimensions, with non-small steps, or with weighted steps. Particular attention will be given to the asymptotic enumeration of such walks using representations of the generating functions as diagonals of rational functions, through the theory of analytic combinatorics in several variables. Several techniques from computational and experimental mathematics will be highlighted, and open conjectures of a probabilistic nature will be discussed.

  • Tuesday October 10, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

    Rigidity of the 3D hierarchical Coulomb gas

    Sourav Chatterjee, Stanford

    The mathematical analysis of Coulomb gases, especially in dimensions higher than one, has been the focus of much recent activity. For the 3D Coulomb, there is a famous prediction of Jancovici, Lebowitz and Manificat that if N is the number of particles falling in a given region, then N has fluctuations of order cube-root of E(N). I will talk about the recent proof of this conjecture for a closely related model, known as the 3D hierarchical Coulomb gas. I will also try to explain, through some toy examples, why such unusually small fluctuations may be expected to appear in interacting gases.

  • Tuesday October 17, 2017 at 15:00, Temple (Wachman Hall 617)

    Homogenization of a class of 1-D nonconvex viscous Hamilton-Jacobi equations with random potential

    Atilla Yilmaz, NYU & Koc University

    There are general homogenization results in all dimensions for (inviscid and viscous) Hamilton-Jacobi equations with random Hamiltonians that are convex in the gradient variable. Removing the convexity assumption has proved to be challenging. There was no progress in this direction until two years ago when the 1-D inviscid case was settled positively and several classes of (mostly inviscid) examples for which homogenization holds were constructed as well as a 2-D inviscid counterexample. Methods that were used in the inviscid case are not applicable to the viscous case due to the presence of the diffusion term.

    In this talk, I will present a new class of 1-D viscous Hamilton-Jacobi equations with nonconvex Hamiltonians for which homogenization holds. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial data have representations involving exponential expectations of controlled Brownian motion in random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in random potential. The proof relies on (i) analyzing the large deviation behavior of the controlled Brownian particle which assumes the role of one of the players in an emergent two-player game, (ii) identifying asymptotically optimal control policies and (iii) constructing correctors which lead to exponential martingales.

    Based on recent joint work with Elena Kosygina and Ofer Zeitouni.

  • Tuesday October 24, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

    Extreme level sets of branching Brownian motion

    Lisa Hartung, NYU

    We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida. (Joint work with A. Cortines, O Louidor.)

  • Tuesday November 7, 2017 at 15:00, 617 Wachman Hall

    Spectrum of random band matrices 

    Indrajit Jana, Temple University

    We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ band matrix of bandwidth $b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law.

  • Tuesday November 14, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

    How fragile are information cascades? 

    Miklos Racz, Berkeley

    It is well known that sequential decision making may lead to information cascades. If the individuals are choosing between a right and a wrong state, and the initial actions are wrong, then the whole cascade will be wrong. We show that if agents occasionally disregard the actions of others and base their action only on their private information, then wrong cascades can be avoided. Moreover, we obtain the optimal asymptotic rate at which the error probability at time t can go to zero. This is joint work with Yuval Peres, Allan Sly, and Izabella Stuhl.

  • Tuesday November 28, 2017 at 15:00, Temple (Wachman Hall 617)

    (Postponed)

    Thomas Leblé, NYU

  • Tuesday December 5, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)

    TBA

    Konstantinos Karatapanis, UPenn

     

Probability Seminars 2018

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

 

  • Tuesday September 4, 2018 at 15:00, Penn (DRL 4C8)

    In between random walk and rotor walk in the square lattice

    Swee Hong Chen, Cornell

    How much randomness is needed to prove a scaling limit result? In this talk we consider this question for a family of random walks on the square lattice. When the randomness is turned to the maximum, we have the symmetric random walk, which is known to scale to a two-dimensional Brownian motion. When the randomness is turned to zero, we have the rotor walk, for which its scaling limit is an open problem. This talk is about random walks that lie in between these two extreme cases and for which we can prove their scaling limit. This is a joint work with Lila Greco, Lionel Levine, and Boyao Li.

  • Tuesday September 11, 2018 at 15:00, Temple (Wachman 617)

    The Sine-beta process: DLR equations and applications

    Thomas Leblé, NYU Courant

    One-dimensional log-gases, or Beta-ensembles, are statistical physics toy models finding their incarnation in random matrix theory. Their limit behavior at microscopic scale is known as the Sine-beta process, its original description involves systems of coupled SDE’s. 
    We give a new description of Sine-beta as an « infinite volume Gibbs measure », using the Dobrushin-Lanford-Ruelle (DLR) formalism, and use it to prove the “rigidity” of the process, in the sense of Ghosh-Peres. If time permits, I will mention another application to the study of fluctuations of linear statistics. Joint work with David Dereudre, Adrien Hardy, and Mylène Maïda.

  • Tuesday September 18, 2018 at 15:00, Temple (Wachman 617)

    Stationary coalescing walks on the lattice

    Arjun Krishnan, University of Rochester

    Consider a measurable dense family of semi-infinite nearest-neighbor paths on the integer lattice in d dimensions. If the measure on the paths is translation invariant, we completely classify their collective behavior in d=2 under mild assumptions. We use our theory to classify the behavior of semi-infinite geodesics in random translation invariant metrics on the lattice; it applies, in particular, to first- and last-passage percolation. We also construct several examples displaying unexpected behaviors. (Joint work with Jon Chaika.)

  • Tuesday September 25, 2018 at 15:00, Penn (DRL 4C8)

    Zeros of polynomials, the distribution of coefficients, and a problem of J.E. Littlewood

    Julian Sahasrabudhe, Cambridge University

    While it is an old and fundamental fact that every (nice enough) even function $f : [-\pi,\pi] \rightarrow \mathbb{C}$ may be uniquely expressed as a cosine series \[ f(\theta) = \sum_{r \geq 0 } C_r\cos(r\theta), \] the relationship between the sequence of coefficients $(C_r)_{r \geq 0 }$ and the behavior of the function $f$ remains mysterious in many aspects. We mention two variations on this theme. First a more probabilistic setting: what can be said about a random variable if we constrain the roots of the probability generating function? We then settle on our main topic; a solution to a problem of J.E. Littlewood about the behavior of the zeros of cosine polynomials with coefficients $C_r \in \{0,1\}$.

  • Tuesday October 2, 2018 at 15:00, Temple (Wachman 617)

    Shifted weights and restricted path length in first-passage percolation

    Firas Rassoul-Agha, University of Utah

    We study standard first-passage percolation via related optimization problems that restrict path length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of geodesic length due to Hammersley, Smythe and Wierman, and Kesten. We study the regularity of the time constant as a function of the shift of weights. For unbounded weights, this function is strictly concave and in case of two or more atoms it has a dense set of singularities. For any weight distribution with an atom at the origin there is a singularity at zero, generalizing a result of Steele and Zhang for Bernoulli FPP. The regularity results are proved by the van den Berg-Kesten modification argument. This is joint work with Arjun Krishnan and Timo Seppäläinen.
     

  • Tuesday October 9, 2018 at 15:00, Temple (Wachman 617)

    The coin turning walk and its scaling limit

    Janos Englander, CU Boulder

    Given a sequence of numbers $p_n ∈ [0, 1]$, consider the following experiment. First, we flip a fair coin and then, at step $n$, we turn the coin over to the other side with probability $p_n, n > 1$, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as $n \to \infty$? We show that a number of phase transitions take place as the turning gets slower (i.e. $p_n$ is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is $p_n = \textrm{const}/n$. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws. The critical regime is particularly interesting: when the corresponding random walk is considered, an interesting process emerges as the scaling limit; also, a connection with Polya urns will be mentioned. This is joint work with S. Volkov (Lund) and Z. Wang (Boulder).

  • Tuesday October 23, 2018 at 15:00, Penn (DRL 4C8)

    Isoperimetric shapes in supercritical bond percolation

    Julian Gold, Northwestern University

    We study the isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ of supercritical bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove a shape theorem for these random graphs, showing that upon rescaling they tend almost surely to a deterministic shape. This limit shape is itself an isoperimetric set for a norm we construct. In addition, we obtain sharp asymptotics for a modification of the Cheeger constant of $\textbf{C}_\infty \cap [-n,n]^d$, settling a conjecture of Benjamini for this modified Cheeger constant. Analogous results are shown for the giant component in dimension two, where we use the original definition of the Cheeger constant, and a more complicated continuum isoperimetric problem emerges as a result.

  • Tuesday October 30, 2018 at 15:00, Temple (Wachman 617)

    Sharp transition of invertibility of sparse random matrices

    Anirban Basak, Tata Institute

    Consider an $n \times n$ matrix with i.i.d. Bernoulli($p$) entries. It is well known that for $p= \Omega(1)$, i.e., $p$ bounded below by some positive constant, the matrix is invertible with high probability. If $p \ll \frac{\log n}{n}$ then the matrix contains zero rows and columns with high probability and hence it is singular with high probability. In this talk, we will discuss the sharp transition of the invertibility of this matrix at $p =\frac{\log n}{n}$. This phenomenon extends to the adjacency matrices of directed and undirected Erdös-Rényi graphs, and random bipartite graphs. Joint work with Mark Rudelson.

  • Tuesday November 6, 2018 at 15:00, Penn (DRL 4C8)

    Applications of CLTs and homogenization for Dyson Brownian Motion to Random Matrix Theory

    Philippe Sosoe, Cornell University

     I will explain how two recent technical developments in Random Matrix Theory allow for a precise description of the fluctuations of single eigenvalues in the spectrum of large symmetric matrices. No prior knowledge of random matrix theory will be assumed. (Based on joint work with B. Landon and H.-T. Yau.)

  • Tuesday November 13, 2018 at 15:00, Penn (DRL 4C8)

    Inference and compression problems on dynamic networks

    Abram Magner, Purdue University

    Networks in the real world are dynamic -- nodes and edges are added and removed over time, and time-varying processes (such as epidemics) run on them. In this talk, I will describe mathematical aspects of some of my recent work with collaborators on statistical inference and compression problems that involve this time-varying aspect of networks. I will focus on two related lines of work: (i) network archaeology -- broadly concerning problems of dynamic graph model validation and inference about previous states of a network given a snapshot of its current state, and (ii) structural compression -- for a given graph model, exhibit an efficient algorithm for invertibly mapping network structures (i.e., graph isomorphism types) to bit strings of minimum expected length. For both classes of problems, I give both information-theoretic limits and efficient algorithms for achieving those limits. Finally, I briefly describe some ongoing projects that continue these lines of work.

  • Tuesday November 27, 2018 at 15:00, Temple (Wachman 617)

    Stick breaking processes, clumping, and Markov chain occupation laws

    Sunder Sethuraman, University of Arizona 

    A GEM (Griffiths-Engen-McCloskey) sequence specifies the (random) proportions in splitting a `resource' infinitely many ways. Such sequences form the backbone of `stick breaking' representations of Dirichlet processes used in nonparametric Bayesian statistics. In this talk, we consider the connections between a class of generalized `stick breaking' processes, an intermediate structure via `clumped' GEM sequences, and the occupation laws of certain time-inhomogeneous Markov chains.

  • Tuesday December 4, 2018 at 15:00, Temple (Wachman 617)

    The algorithmic hardness threshold for continuous random energy models

    Pascal Maillard, Orsay/CRM

    I will report on recent work with Louigi Addario-Berry on algorithmic hardness for finding low-energy states in the continuous random energy model of Bovier and Kurkova. This model can be regarded as a toy model for strongly correlated random energy landscapes such as the Sherrington-Kirkpatrick model. We exhibit a precise and explicit hardness threshold: finding states of energy above the threshold can be done in linear time, while below the threshold this takes exponential time for any algorithm with high probability. I further discuss what insights this yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.

Probability Seminars 2019

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

 

  • Tuesday January 22, 2019 at 15:00, Penn (DRL 4C8)

    One-point function estimates and natural parametrization for loop-erased random walk in three dimensions

    Xinyi Li, University of Chicago

    In this talk, I will talk about loop-erased random walk (LERW) in three dimensions. I will first give an asymptotic estimate on the probability that 3D LERW passes a given point (commonly referred to as the one-point function). I will then talk about how to apply this estimate to show that 3D LERW as a curve converges to its scaling limit in natural parametrization. If time permits, I will also talk about the asymptotics of non-intersection probabilities of 3D LERW with simple random walk. This is a joint work with Daisuke Shiraishi (Kyoto).

  • Tuesday January 29, 2019 at 15:00, Temple (Wachman 617)

    Fractional Gaussian fields in geometric quantization and the semi-classical analysis of coherent states

    Alexander Moll, Northeastern University

    The Born Rule (1926) formalized in von Neumann's spectral theorem (1932) gives a precise definition of the random outcomes of quantum measurements as random variables from the spectral theory of non-random matrices. In [M. 2017], the Born rule provided a way to derive limit shapes and global fractional Gaussian field fluctuations for a large class of point processes from the first principles of geometric quantization and semi-classical analysis of coherent states. Rather than take a point process as a starting point, these point process are realized as auxiliary objects in an analysis that starts instead from a classical Hamiltonian system with possibly infinitely-many degrees of freedom that is not necessarily Liouville integrable. In this talk, we present these results with a focus on the case of one degree of freedom, where the core ideas in the arguments are faithfully represented.

  • Tuesday February 5, 2019 at 15:00, Penn (DRL 4C8)

    Conformal embedding and percolation on the uniform triangulation

    Xin Sun, Columbia University

    Following Smirnov’s proof of Cardy’s formula and Schramm’s discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice has been achieved. Smirnov’s proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk, I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for critical percolation on uniform triangulations. I will also explain how this result fits in the larger picture of random planar maps and Liouville quantum gravity.

  • Tuesday February 19, 2019 at 15:00, Penn (DRL 4C8)

    Asymptotic zero distribution of random polynomials

    Duncan Dauvergne, University of Toronto

    It is well known that the roots of a random polynomial with i.i.d. coefficients tend to concentrate near the unit circle. In particular, the zero measures of such random polynomials converge almost surely to normalized Lebesgue measure on the unit circle if and only if the underlying coefficient distribution satisfies a particular moment condition. In this talk, I will discuss how to generalize this result to random sums of orthogonal (or asymptotically minimal) polynomials.

  • Tuesday February 26, 2019 at 15:00, Penn (DRL 4C8)

    Distances between random orthogonal matrices and independent normals

    Tiefeng Jiang, University of Minnesota

    We study the distance between Haar-orthogonal matrices and independent normal random variables. The distance is measured by the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. Optimal rates are obtained. This is a joint work with Yutao Ma.

  • Tuesday March 19, 2019 at 15:00, Penn (DRL 4C8)

    Delocalization of random band matrices

    Fan Yang, UCLA

    We consider Hermitian random band matrices $H$ in dimension $d$, where the entries $h_{xy}$, indexed by $x,y \in [1,N]^d$, vanish if $|x-y|$ exceeds the band width $W$. It is conjectured that a sharp transition of the eigenvalue and eigenvector statistics occurs at a critical band width $W_c$, with $W_c=\sqrt{N}$ in $d=1$, $W_c=\sqrt{\log N}$ in $d=2$, and $W_c=O(1)$ in $d\ge 3$. Recently, Bourgade, Yau and Yin proved the eigenvector delocalization for 1D random band matrices with generally distributed entries and band width $W\gg N^{3/4}$. In this talk, we will show that for $d\ge 2$, the delocalization of eigenvectors in a certain averaged sense holds under the condition $W\gg N^{2/(2+d)}$. Based on joint work with Bourgade, Yau and Yin.

  • Tuesday March 26, 2019 at 15:00, Penn (DRL 4C8)

    Large deviations for functionals of Gaussian processes

    Xiaoming Song, Drexel University

    We prove large deviation principles for $\int_0^t \gamma(X_s)ds$, where $X$ is a $d$-dimensional Gaussian process and $\gamma(x)$ takes the form of the Dirac delta function $\delta(x)$, $|x|^{-\beta}$ with $\beta\in (0,d)$, or $\prod_{i=1}^d |x_i|^{-\beta_i}$ with $\beta_i\in(0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.

  • Tuesday April 2, 2019 at 15:00, Temple (Wachman 617)

    Geometry of the corner growth model

    Timo Seppalainen, UW-Madison

    The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah). 

  • Tuesday April 9, 2019 at 15:00, Temple (Wachman 617)

    Eigenvectors of non-Hermitian random matrices

    Guillaume Dubach, Courant Institute, NYU

    Eigenvectors of non-Hermitian matrices are non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables also quantify the stability of the spectrum, and characterize the joint eigenvalue increments under Dyson-type dynamics. Overlaps first appeared in the physics literature, when Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results are expected to hold in other integrable models, and some have been established for quaternionic Gaussian matrices.

  • Tuesday April 16, 2019 at 15:00, Temple (Wachman 617)

    Stochastic homogenization for reaction-diffusion equations

    Jessica Lin, McGill University

    I will present several results concerning the stochastic homogenization for reaction-diffusion equations. We consider reaction-diffusion equations with nonlinear, heterogeneous, stationary-ergodic reaction terms. Under certain hypotheses on the environment, we show that the typical large-time, large-scale behavior of solutions is governed by a deterministic front propagation. Our arguments rely on analyzing a suitable analogue of “first passage times” for solutions of reaction-diffusion equations. In particular, under these hypotheses, solutions of heterogeneous reaction-diffusion equations with front-like initial data become asymptotically front-like with a deterministic speed. This talk is based on joint work with Andrej Zlatos.

  • Tuesday April 30, 2019 at 15:00, Temple (Wachman 617)

    The geometry of the last passage percolation problem

    Tom Alberts, University of Utah

    Last passage percolation is a well-studied model in probability theory that is simple to state but notoriously difficult to analyze. In recent years it has been shown to be related to many seemingly unrelated things: longest increasing subsequences in random permutations, eigenvalues of random matrices, and long-time asymptotics of solutions to stochastic partial differential equations. Much of the previous analysis of the last passage model has been made possible through connections with representation theory of the symmetric group that comes about for certain exact choices of the random input into the last passage model. This has the disadvantage that if the random inputs are modified even slightly then the analysis falls apart. In an attempt to generalize beyond exact analysis, recently my collaborator Eric Cator (Radboud University, Nijmegen) and I have started using tools of tropical geometry to analyze the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model and our work to use it to produce probabilistic information.

  • Tuesday September 3, 2019 at 15:00, Penn (DRL 4C8)

    Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$

    Ewain Gwynne, University of Cambridge

    We show that for each $\gamma \in (0,2)$, there is a unique metric associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the Gaussian free field $h$ on a planar domain $U$, there is a unique random metric $D_h =$ "$e^{\gamma h} (dx^2 + dy^2)$" on $U$ which is uniquely characterized by a list of natural axioms.

    The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the Gaussian free field. Earlier work by Ding, Dubédat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. We show that the subsequential limit is unique and satisfies our list of axioms. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield.

    Based on four joint papers with Jason Miller, one joint paper with Julien Dubédat, Hugo Falconet, Josh Pfeffer, and Xin Sun, and one joint paper with Josh Pfeffer.
     

  • Tuesday September 10, 2019 at 15:00, Temple (Wachman 617)

    Semigroups for one-dimensional Schrödinger operators with multiplicative white noise

    Pierre Yves Gaudreau Lamarre, Princeton University

    In this talk, we are interested in the semigroup theory of continuous one-dimensional random Schrödinger operators with white noise. We will begin with a brief reminder of the rigorous definition of these operators as well as some of the problems in which they naturally arise. Then, we will discuss the proof of a Feynman-Kac formula describing their semigroups. In closing, we will showcase an application of this new semigroup theory to the study of rigidity (in the sense of Ghosh-Peres) of random Schrödinger eigenvalue point processes.

    Some of the results discussed in this talk are joint work with Promit Ghosal (Columbia) and Yuchen Liao (Michigan).
     

  • Tuesday September 17, 2019 at 15:00, Penn (DRL 4C8)

    Hard-core models in discrete 2D

    Izabella Stuhl, Pennsylvania State University

    Do hard disks in the plane admit a unique Gibbs measure at high density? This is one of the outstanding open problems of statistical mechanics, and it seems natural to approach it by requiring the centers to lie in a fine lattice; equivalently, we may fix the lattice, but let the Euclidean diameter $D$ of the hard disks tend to infinity. Unlike most models in statistical physics, we find non-universality and connections to number theory, with different new phenomena arising in the triangular lattice $\mathbb{A}_2$, the square lattice $\mathbb{Z}^2$ and the hexagonal tiling $\mathbb{H}_2$.

    In particular, number-theoretic properties of the exclusion diameter $D$ turn out to be important. We analyze high-density hard-core Gibbs measures via Pirogov-Sinai theory. The first step is to identify periodic ground states, i.e., maximal density disk configurations which cannot be locally 'improved'. A key finding is that only certain 'dominant' ground states, which we determine, generate nearby Gibbs measures. Another important ingredient is the Peierls bound separating ground states from other admissible configurations.

    Answers are provided in terms of Eisenstein primes for $\mathbb{A}_2$ and norm equations in the ring $\mathbb{Z}[\sqrt{3}]$ for $\mathbb{Z}^2$. The number of high-density hard-core Gibbs measures grows indefinitely with $D$ but non-monotonically. In $\mathbb{Z}^2$ we analyze the phenomenon of 'sliding' and show it is rare.

    This is a joint work with A. Mazel and Y. Suhov.
     

  • Tuesday September 24, 2019 at 15:00, Temple (Wachman 617)

    Stationary dynamics in finite time for the totally asymmetric simple exclusion process

    Axel Saenz, University of Virginia

    The totally asymmetric simple exclusion process (TASEP) is a Markov process that is the prototypical model for transport phenomena in non-equilibrium statistical mechanics. It was first introduced by Spitzer in 1970, and in the last 20 years, it has gained a strong resurgence in the emerging field of "Integrable Probability" due to exact formulas from Johansson in 2000 and Tracy and Widom in 2007 (among other related formulas and results). In particular, these formulas led to great insights regarding fluctuations related to the Tracy-Widom distribution and scalings to the Kardar-Parisi-Zhang (KPZ) stochastic differential equation. 

    In this joint work with Leonid Petrov (University of Virginia), we introduce a new and simple Markov process that maps the distribution of the TASEP at time $t >0$ , given step initial time data, to the distribution of the TASEP at some earlier time $t-s>0$. This process "back in time" is closely related to the Hammersley process introduced by Hammersley in 1972, which later found a resurgence in the longest increasing subsequence problem in the work of Aldous and Diaconis in 1995. Hence, we call our process the backwards Hammersley-type process (BHP). As a fun application of our results, we have a new proof of the limit shape for the TASEP. The central objects in our constructions and proofs are the Schur point processes and the Yang-Baxter equation for the $sl_2$ quantum affine Lie algebra. In this talk, we will discuss the background in more detail and will explain the main ideas behind the constructions and proof. 
     

  • Tuesday October 1, 2019 at 15:00, Penn (DRL 4C8)

    Dynamics for spherical spin glasses: Disorder dependent initial conditions

    Amir Dembo, Stanford University

    In this talk, based on a joint work with Eliran Subag, I will explain how to rigorously derive the integro-differential equations that arise in the thermodynamic limit of the empirical correlation and response functions for Langevin dynamics in mixed spherical p-spin disordered mean-field models.

    I will then compare the large-time asymptotic of these equations in case of a uniform (infinite-temperature)  starting point, to what one obtains when starting within one of the spherical bands on which the Gibbs measure concentrates at low temperature, commenting on the existence of an aging phenomenon, and on the relations with the recently discovered geometric structure of the Gibbs measures at low temperature.
     

  • Tuesday October 8, 2019 at 15:00, Temple (Wachman 617)

    Lower-tail large deviations of the KPZ equation

    Li-Cheng Tsai, Rutgers University

    Consider the solution of the KPZ equation with the narrow wedge initial condition. We prove the one-point, lower-tail Large Deviation Principle (LDP) of the solution, with time $t\to\infty$ being the scaling parameter, and with an explicit rate function. This result confirms existing physics predictions. We utilize a formula from Borodin and Gorin (2016) to convert the LDP of the KPZ equation to calculating an exponential moment of the Airy point process, and analyze the latter via the stochastic Airy operator and Riccati transform.
     

  • Tuesday October 15, 2019 at 15:00, Penn (DRL 4C8)

    Local regime of random band matrices

    Tatyana Shcherbina, Princeton University

    Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.
     

  • Tuesday October 29, 2019 at 15:00, Penn (DRL 4C8)

    Geometric TAP approach for spherical spin glasses

    Eliran Subag, Courant Institute, NYU

    The celebrated Thouless-Anderson-Palmer approach suggests a way to relate the free energy of a mean-field spin glass model to the solutions of certain self-consistency equations for the local magnetizations. In this talk I will first describe a new geometric approach to define free energy landscapes for general spherical mixed p-spin models and derive from them a generalized TAP representation for the free energy. I will then explain how these landscapes are related to various concepts and problems: the pure states decomposition, ultrametricity, temperature chaos, and optimization of full-RSB models.
     

  • Tuesday November 5, 2019 at 15:00, Temple (Wachman 617)

    Absence of backward infinite paths in first-passage percolation in arbitrary dimension

    Michael Damron, Georgia Tech

    In first-passage percolation (FPP), one places weights $(t_e)$ on the edges of $\mathbb{Z}^d$ and considers the induced metric. Optimizing paths for this metric are called geodesics, and infinite geodesics are infinite paths all whose finite subpaths are geodesics. It is a major open problem to show that in two dimensions, with i.i.d. continuous weights, there are no bigeodesics (doubly-infinite geodesics). In this talk, I will describe work on bigeodesics in arbitrary dimension using "geodesic graph'' measures introduced in '13 in joint work with J. Hanson. Our main result is that these measures are supported on graphs with no doubly-infinite paths, and this implies that bigeodesics cannot be constructed in a translation-invariant manner in any dimension as limits of point-to-hyperplane geodesics. Because all previous works on bigeodesics were for two dimensions and heavily used planarity and coalescence, we must develop new tools based on the mass transport principle. Joint with G. Brito (Georgia Tech) and J. Hanson (CUNY).
     

  • Tuesday November 12, 2019 at 15:00, Penn (DRL 4C8)

    Sharp threshold for the Ising perceptron model

    Changji Xu, University of Chicago

    Consider the discrete cube $\{-1,1\}^N$ and a random collection of half spaces which includes each half space $H(x) := \{ y \in \{-1,1\}^N: x \cdot y \geq \kappa \sqrt{N} \}$ for $x \in \{-1,1\}^N$ independently with probability $p$. Is the intersection of these half spaces empty? This is called the Ising perceptron model under Bernoulli disorder. We prove that this event has a sharp threshold, that is, the probability that the intersection is empty increases quickly from $\epsilon$ to $1- \epsilon$ when $p$ increases only by a factor of $1 + o(1)$ as $N \to \infty$.

  • Tuesday November 19, 2019 at 15:00, Temple (Wachman 617)

    The Edwards-Wilkinson limit of the KPZ equation in $d>1$

    Yu Gu, Carnegie Mellon University

    In this talk, I will explain some recent work where we prove that in a certain weak disorder regime, the KPZ equation scales to the Edwards-Wilkinson equation in $d>1$.

  • Tuesday December 3, 2019 at 15:00, Penn (DRL 4C8)

    Maximum of 3D Ising interfaces

    Eyal Lubetzky, Courant Institute, NYU

    Consider the random surface separating the plus and minus phases, above and below the $xy$-plane, in the low temperature Ising model in dimension $d\geq 3$. Dobrushin (1972) showed that if the inverse-temperature $\beta$ is large enough then this interface is localized: it has $O(1)$ height fluctuations above a fixed point, and its maximum height on a box of side length $n$ is $O_P ( \log n )$.

    We study the large deviations of the interface in Dobrushin’s setting, and derive a shape theorem for its "pillars" conditionally on reaching an atypically large height. We use this to obtain a law of large numbers for the maximum height $M_n$ of the interface: $M_n/ \log n$ converges to $c_\beta$ in probability, where $c_\beta$ is given by a large deviation rate in infinite volume. Furthermore, the sequence $(M_n - E[M_n])_{n\geq 1}$ is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tail bounds.

    Joint work with Reza Gheissari.

Probability Seminars 2020

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

 

  • Tuesday January 21, 2020 at 15:00, Penn (DRL 4C8)

    Fast randomized iterative numerical linear algebra for quantum chemistry (and other applications)

    Jonathan Weare, Courant Institute, NYU

    I will discuss a family of recently developed stochastic techniques for linear algebra problems involving very large matrices.  These methods can be used to, for example, solve linear systems, estimate eigenvalues/vectors, and apply a matrix exponential to a vector, even in cases where the desired solution vector is too large to store.  The first incarnations of this idea appear for dominant eigenproblems arising in statistical physics and in quantum chemistry and were inspired by the real space diffusion Monte Carlo algorithm which has been used to compute chemical ground states since the 1970's.  I will discuss our own general framework for fast randomized iterative linear algebra as well share a very partial explanation for their effectiveness.  I will also report on the progress of an ongoing collaboration aimed at developing fast randomized iterative schemes specifically for applications in quantum chemistry.  This talk is based on joint work with Lek-Heng Lim, Timothy Berkelbach, Sam Greene, and Rob Webber.

  • Tuesday January 28, 2020 at 15:00, Temple (Wachman 617)

    The KPZ fixed point

    Konstantin Matetski, Columbia University

    The KPZ universality class is a broad collection of models, which includes directed random polymers, interacting particle systems and random interface growth, characterized by unusual scale of fluctuations which also appear in the random matrix theory. The KPZ fixed point is a scaling invariant Markov process which is the conjectural universal limit of all models in the class. A complete description of the KPZ fixed point was obtained in a joint work with Jeremy Quastel and Daniel Remenik. In this talk I will describe how the KPZ fixed point was derived by solving a special model in the class called TASEP.

  • Tuesday February 4, 2020 at 15:00, Penn (DRL 4C8)

    Scaling limit of a directed polymer among a Poisson field of independent walks

    Jian Song, Shandong University

    We consider a directed polymer model in dimension $1+1$, where the disorder is given by the occupation field of a Poisson system of independent random walks on $\mathbb{Z}$. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation with a Gaussian noise whose space-time covariance is given by the heat kernel.

  • Tuesday February 11, 2020 at 15:00, Temple (Wachman 617)

    Entropy of ribbon tilings

    Vladislav Kargin, Binghamton University

    I will talk about ribbon tilings, which have been originally introduced and studied by Pak and Sheffield. These are a generalization of the domino tilings which, unfortunately, lacks relations to determinants and spanning trees but still retains some of the nice properties of domino tilings. I will explain how ribbon tilings are connected to multidimensional heights and acyclic orientations, and present some results about enumeration of these tilings on simple regions. Joint work with Yinsong Chen.

  • Tuesday February 18, 2020 at 15:00, Penn (DRL 4C8)

    Extreme eigenvalue distributions of sparse random graphs

    Jiaoyang Huang, Institute for Advanced Study

    I will discuss the extreme eigenvalue distributions of adjacency matrices of sparse random graphs, in particular the Erdős-Rényi graphs $G(N,p)$ and the random $d$-regular graphs. For Erdős-Rényi graphs, there is a crossover in the behavior of the extreme eigenvalues. When the average degree $Np$ is much larger than $N^{1/3}$, the extreme eigenvalues have asymptotically Tracy-Widom fluctuations, the same as Gaussian orthogonal ensemble. However, when $N^{2/9}\ll Np\ll N^{1/3}$ the extreme eigenvalues have asymptotically Gaussian fluctuations. The extreme eigenvalues of random $d$-regular graphs are more rigid, we prove on the regime $N^{2/9}\ll d\ll N^{1/3}$ the extremal eigenvalues are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by the Tracy-Widom statistics. Thus, in the same regime of $d$, $52\%$ of all $d$-regular graphs have the second-largest eigenvalue strictly less than $2\sqrt{d-1}$. These are based on joint works with Roland Bauerschmids, Antti Knowles, Benjamin Landon and Horng-Tzer Yau.

  • Tuesday February 25, 2020 at 15:00, Temple (Wachman 617)

    Universality of extremal eigenvalue statistics of random matrices

    Benjamin Landon, MIT

    The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory.  However, the behavior of certain "extremal" or "critical" observables is not fully understood.  Towards the former, we discuss progress on the universality of the largest gap between consecutive eigenvalues.  With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.

  • Tuesday March 3, 2020 at 15:00, Penn (DRL 4C8)

    Estimation of Wasserstein distances in the spiked transport model

    Jonathan Niles-Weed, Courant Institute, NYU

    We propose a new statistical model, generalizing the spiked covariance model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study various probabilistic and statistical features of this model, including the estimation of the Wasserstein distance, which we show can be accomplished by an estimator which avoids the "curse of dimensionality" typically present in high-dimensional problems involving the Wasserstein distance. However, this estimator does not seem possible to compute in polynomial time, and we give evidence that any computationally efficient estimator is bound to suffer from the curse of dimensionality. Our results therefore suggest the existence of a computational-statistical gap.
     

    Joint work with Philippe Rigollet.

     

  • Tuesday March 17, 2020 at 15:00, Penn (DRL 4C8)

    (POSTPONED)

    Robert Hough, Stony Brook University

     

  • Tuesday March 24, 2020 at 15:00, Wachman 617

    (POSTPONED)

    Kavita Ramanan, Brown University

     

  • Tuesday March 31, 2020 at 15:00, Penn (DRL 4C8)

    (POSTPONED)

    Yilin Wang, MIT

     

  • Tuesday April 7, 2020 at 15:00, Temple (Wachman 617)

    (POSTPONED)

    Hoi Nguyen, Ohio State University

     

  • Tuesday April 14, 2020 at 15:00, Penn (DRL 4C8)

    (POSTPONED)

    Louis Fan, Indiana University

     

  • Tuesday April 21, 2020 at 15:00, Temple (Wachman 617)

    (POSTPONED)

    Kyle Luh, Harvard University

     

  • Tuesday April 28, 2020 at 15:00, Penn (DRL 4C8)

    (POSTPONED)

    Gourab Ray, University of Victoria

     

Probability Seminars 2021

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

 

  • Tuesday September 7, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

    Delocalization and quantum diffusion of random band matrices in high dimensions

    Fan Yang, UPenn

    We consider a Hermitian random band matrix $H$ on the $d$-dimensional lattice of linear size $L$. Its entries are independent centered complex Gaussian random variables with variances $s_{xy}$, that are negligible if $|x-y|$ exceeds the band width $W$. In dimensions eight or higher, we prove that, as long as $W > L^\epsilon$ for a small constant $\epsilon>0$, with high probability, most bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to $L$. Moreover, we also prove a quantum diffusion result of this model in terms of the Green's function of $H$. Joint work with Horng-Tzer Yau and Jun Yin.

  • Tuesday September 14, 2021 at 15:30, Temple (Wachman Hall 617)

    Spanning clusters and subcritical connectivity in high-dimensional percolation

    Jack Hanson, City College & The Graduate Center, CUNY

    In their study of percolation, physicists have proposed "scaling hypotheses" relating the behavior of the model in the critical ($p = p_c$) and subcritical ($p < p_c$) regimes. We show a version of such a scaling hypothesis for the one-arm probability $\pi(n;p)$ — the probability that the open cluster of the origin has Euclidean diameter at least $n$.

    As a consequence of our analysis, we obtain the correct scaling of the lower tail of cluster volumes and the chemical (intrinsic) distances within clusters. We also show that the number of spanning clusters of a side-length $n$ box is tight on scale $n^{d-6}$. A new tool of our analysis is a sharp asymptotic for connectivity probabilities when paths are restricted to lie in half-spaces.

  • Tuesday September 21, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

    Hamilton-Jacobi equations for statistical inference problems

    Jiaming Xia, UPenn

    In this talk, I will first present the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. We compare the limit with the solution to a certain Hamilton-Jacobi equation, following the recent approach by Jean-Christophe Mourrat. The motivation comes from the averaged free energy solving an approximate Hamilton-Jacobi equation. We consider two notions of solutions which are weak solutions and viscosity solutions. The two types of solutions require different treatments and each has its own advantages. At the end of this part, I will show an example of application of our results to a model with i.i.d. entries and symmetric interactions. If time permits, I will talk about the same problem but with a different model, namely, the multi-layer generalized linear model. I will mainly explain the iteration method as an important tool used in our proof. This is based on joint works with Hong-Bin Chen and J.-C. Mourrat, NYU.

  • Tuesday September 28, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

    Wilson loop expectations as sums over surfaces in 2D

    Minjae Park, MIT

    Although lattice Yang-Mills theory on $\mathbb{Z}^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on $\mathbb{R}^d$ is a major open problem when $d\ge 3$. Such a theory should assign a Wilson loop expectation to each suitable collection $\mathcal{L}$ of loops in $\mathbb{R}^d$. One classical approach is to try to represent this expectation as a sum over surfaces with boundary $\mathcal{L}$. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.

    In this talk, we show how to make sense of Yang-Mills integrals as surface sums for $d=2$, where the continuum theory is already understood. We also obtain an alternative proof of the Makeenko-Migdal equation and a version of the Gross-Taylor expansion. Joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu. 
     

     

  • Tuesday October 5, 2021 at 15:30, Temple (Wachman Hall 617)

    Singularities in the spectrum of random block matrices

    David Renfrew, SUNY Binghamton

    We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.

  • Tuesday October 12, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

    The local limit theorem on nilpotent groups

    Robert Hough, Stony Brook University

    Alexopoulos proved local limit theorems for measures with a density and lattice measures in the general setting of groups of moderate growth. On the Heisenberg group, Breuillard's thesis obtained a local limit theorem for general measures subject to a condition on the characteristic function, and asked if this condition can be removed. I will discuss two new local limit theorems, one joint with Diaconis, that treat local limit theorems on nilpotent Lie groups driven by general measures. We prove Breuillard's conjecture and also solve a problem of Diaconis and Saloff-Coste on the mixing of the central coordinate in unipotent matrix walks modulo $p$. 

  • Tuesday October 19, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

    Integrability of boundary Liouville CFT

    Guillaume Remy, Columbia University

    Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced in physics by A. Polyakov to describe a canonical random 2d surface. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will study the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its correlation functions. Our latest result is derived using conformal welding of random surfaces, in relation with the Schramm-Loewner evolutions. We will also discuss the connection with the conformal blocks of CFT which are fundamental functions determined by conformal invariance that underlie the exact solvability of CFT. Based on joint works with Morris Ang, Promit Ghosal, Xin Sun, Yi Sun and Tunan Zhu.

  • Tuesday October 26, 2021 at 15:30, Temple (Wachman Hall 617)

    Gaussian, stable, and tempered stable limiting distributions for random walks in cooling random environments

    Jonathon Peterson, Purdue University

    Random walks in cooling random environments are a model of random walks in dynamic random environments where the random environment is re-sampled at a fixed sequence of times (called the cooling sequence) and the environment remains constant between these re-sampling times. We study the limiting distributions of the walk in the case when distribution on environments is such that a walk in a fixed environment has an $s$-stable limiting distribution for some $s \in (1,2)$. It was previously conjectured that for cooling maps whose gaps between re-sampling times grow polynomially that the model should exhibit a phase transition from Gaussian limits to $s$-stable depending on the exponent of the polynomial growth of the re-sampling gaps. We confirm this conjecture, identifying the precise exponent at which the phase transition occurs and proving that at the critical exponent the limiting distribution is a generalized tempered $s$-stable distribution. The proofs require us to prove some previously unknown facts about one-dimensional random walks in random environments which are of independent interest. 

  • Tuesday November 2, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

    Loewner chains driven by complex Brownian motion

    Joshua Pfeffer, Columbia University

    In my talk I will discuss Loewner chains whose driving functions are complex Brownian motions with general covariance matrices.  This extends the notion of Schramm-Loewner evolution (SLE) by allowing the driving function to be complex-valued and not just real-valued.  We show that these Loewner chains exhibit the same phases (simple, swallowing, and space-filling) as SLE, and we explicitly characterize the values of the covariance matrix corresponding to each phase.  In contrast to SLE, we show that the evolving left hulls are a.s. not generated by curves, and that they a.s. disconnect each fixed point in the plane from infinity before absorbing the point.

    This talk is based on a joint work with Ewain Gwynne. 

  • Tuesday November 9, 2021 at 15:30, Temple (Wachman Hall 617)

    On the limiting shape of Young diagram associated with Markov random words

    Christian Houdré, Georgia Tech

    Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain, with state space a totally ordered finite alphabet of size $m$. Using combinatorial constructions and weak invariance principles, we obtain the limiting shape of the associated RSK Young diagrams as a multidimensional Brownian functional. Since the length of the top row of the Young diagrams is also the length of the longest weakly increasing subsequences of $(X_k)_{1\le k \le n}$, the corresponding limiting law follows. We relate our results to a conjecture of Kuperberg by providing, under a cyclic condition, a spectral characterization of the Markov transition matrix precisely characterizing when the limiting shape is the spectrum of the $m \times m$ traceless GUE. For each $m \ge 4$, this characterization identifies a proper, non-trivial class of cyclic transition matrices producing such a limiting shape. However, for $m=3$, all cyclic Markov chains have such a limiting shape, a fact previously only known for $m=2$. For $m$ arbitrary, we also study reversible Markov chains and obtain a characterization of symmetric Markov chains for which the limiting shape is the spectrum of the traceless GUE. To finish, we explore, in this general setting, connections between various limiting laws and spectra of Gaussian random matrices, focusing in particular on the relationship between the terminal points of the Brownian motions, the diagonal terms of the random matrix, and the scaling of its off-diagonal terms, a scaling we conjecture to be a function of the spectrum of the covariance matrix governing the Brownian motion.

    Joint work with Trevis Litherland.

  • Tuesday November 16, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

    The skew Brownian permuton

    Jacopo Borga, Stanford University

    Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permutons. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, mainly focusing on the case of pattern-avoiding permutations. The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian permutons. This family includes (as particular cases) some already studied limiting permutons, such as the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families of random constrained permutations converge to some new instances of skew Brownian permutons. The construction of these new limiting objects will lead us to investigate an intriguing connection with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions. If time permits, we will present some conjectures on how it should be possible to construct these new limiting permutons directly from the Liouville quantum gravity decorated with two SLE curves. 

  • Tuesday November 30, 2021 at 15:30, Temple (Wachman Hall 617)

    Central limit theorem for the characteristic polynomial of general beta-ensembles

    Krishnan Mody, Courant Institute, NYU

    I will discuss recent work with P. Bourgade and M. Pain in which we show that the log-characteristic polynomial for general beta ensembles converges to a log-correlated field in the large-dimension limit. The proof of this result relies on a so-called optimal local law, which I will explain and prove in the Gaussian case. I will explain how the local law is useful, and give an outline of the proof of the log-correlated field.

  • Tuesday December 7, 2021 at 15:30, Temple (Wachman Hall 617)

    Large values of the Riemann zeta function in short intervals

    Louis-Pierre Arguin, Baruch College & The Graduate Center, CUNY

    I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function in small intervals of the critical line. This problem has interesting connections with the extreme value statistics of IID and log-correlated random variables, as well as random matrix theory.

Probability Seminars 2022

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

 

  • Tuesday January 25, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Strong Quantum Unique Ergodicity and its Gaussian fluctuations for Wigner matrices

    Giorgio Cipolloni, Princeton University
     

    We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis, which is a strong form of Quantum Unique Ergodicity (QUE) with optimal speed of convergence. Then, using this a priori bound as an input, we analyze the Stochastic Eigenstate Equation (SEE) and prove Gaussian fluctuations in the QUE.

    The main methods behind the above results are:

    (i) multi-resolvent local laws established via a novel bootstrap scheme;

    (ii) energy estimates for SEE.

     

  • Tuesday February 1, 2022 at 15:30, Temple (Wachman Hall 617)

    On convergence of the cavity and Bolthausen’s TAP iterations to the local magnetization

    Si Tang, Lehigh University

    The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington-Kirkpatrick model. In the seminal work, Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida-Thouless transition line. However, it was unclear if this asymptotic solution coincides with the local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called Approximate Message Passing algorithm, a generalization of Bolthausen’s iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen’s scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated. This is a joint work with Wei-Kuo Chen (University of Minnesota).

  • Tuesday February 8, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Lozenge tilings and the Gaussian free field on a cylinder

    Marianna Russkikh, MIT

    We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes. 

  • Tuesday February 15, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Scaling limits of the Laguerre unitary ensemble

    Xuan Wu, University of Chicago

     

    In this talk, we will discuss the LUE, focusing on the scaling limits. On the hard-edge side, we construct the $\alpha$-Bessel line ensemble for all $\alpha \in \mathbb{N}_0$. This novel Gibbsian line ensemble enjoys the $\alpha$-squared Bessel Gibbs property. Moreover, all $\alpha$-Bessel line ensembles can be naturally coupled together in a Bessel field, which enjoys rich integrable structures. We will also talk about work in progress on the soft-edge side, where we expect to have the Airy field as the scaling limit. This talk is based on joint works with Lucas Benigni, Pei-Ken Hung, and Greg Lawler.

     

  • Tuesday February 22, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    The local environment of a geodesic in Last-Passage Percolation

    Lingfu Zhang, Princeton University
     

    In exponential Last-Passage Percolation, each vertex in the 2D lattice is assigned an i.i.d. exponential weight, and the geodesic between a pair of vertices refers to the up-right path connecting them, with the maximum total weight along the path. This model was first introduced to model fluid flow through a random medium. It is also a central model in the KPZ universality class and related to various natural processes.

    A classical question asks what a geodesic looks like locally, and how weights on and nearby the geodesic behave. In this talk, I will present new results on the convergence of the ‘environment’ as seen from a typical point along the geodesic, and convergence of the corresponding empirical measure. In addition, we obtain an explicit description of the limiting ‘environment’. This in principle enables one to compute all the local statistics of the geodesic, and I will talk about some surprising and interesting examples.

    This is based on joint work with James Martin and Allan Sly.

     

  • Tuesday March 8, 2022 at 15:30, Temple (Wachman Hall 617)

    Large deviation estimates for Selberg’s central limit theorem and applications

    Emma Bailey, The Graduate Center, CUNY

    Selberg’s celebrated central limit theorem shows that the logarithm of the zeta function at a typical point on the critical line behaves like a complex, centered Gaussian random variable with variance $\log\log T$. This talk will present recent results showing that the Gaussian decay persists in the large deviation regime, at a level on the order of the variance, improving on the best known bounds in that range.  We also present various applications, including on the maximum of the zeta function in short intervals. Whilst the results are number theoretic, the tools used are predominantly probabilistic in nature.  This work is joint with Louis-Pierre Arguin. 

  • Tuesday March 15, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Metric growth dynamics in Liouville quantum gravity

    Hugo Falconet, Courant Institute, NYU

    Liouville quantum gravity (LQG) is a canonical model of random geometry. Associated with the planar Gaussian free field, this geometry with conformal symmetries was introduced in the physics literature by Polyakov in the 80’s and is conjectured to describe the scaling limit of random planar maps. In this talk, I will introduce LQG as a metric measure space and discuss recent results on the associated metric growth dynamics. The primary focus will be on the dynamics of the trace of the free field on the boundary of growing LQG balls.

    Based on a joint work with Julien Dubédat.

  • Tuesday March 22, 2022 at 15:30, Temple (Wachman Hall 617)

    From generalized Ray-Knight theorems to functional limit theorems for some models of self-interacting random walks on integers

    Elena Kosygina, Baruch College & The Graduate Center, CUNY

    For several models of self-interacting random walks (SIRWs), generalized Ray-Knight theorems for edge local times are a very useful tool for studying the limiting distributions of these walks. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss recent results (joint work with Thomas Mountford, EPFL, and Jon Peterson, Purdue University) which resolve an open question posed in Toth’s paper. We show that, in the asymptotically free case, the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth), while in the polynomially self-repelling case, the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of perturbed Brownian motions. This negative result was somewhat unexpected. Conjectures on whether there is a suitable limiting process in this case and, if yes, what it might be are welcome.

  • Tuesday March 29, 2022 at 15:30, Temple (Wachman Hall 617)

    Localization and delocalization in Erdős–Rényi graphs

    Johannes Alt, Courant Institute, NYU

    We consider the Erdős–Rényi graph on N vertices with edge probability d/N. It is well known that the structure of this graph changes drastically when d is of order log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of the eigenvectors remains delocalized. In this talk, I will present the phase diagram depicting these localized and delocalized phases and our recent progress in establishing it rigorously.

    This is based on joint works with Raphael Ducatez and Antti Knowles.

  • Tuesday April 5, 2022 at 15:30, Temple (Wachman Hall 617)

    Uniqueness in Cauchy problems for diffusive real-valued strict local martingales

    Kasper Larsen, Rutgers University

    For a real-valued one-dimensional diffusive strict local martingale, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution. We exemplify our results using quadratic normal volatility models and the two-dimensional Bessel process. Joint work with Umut Cetin (LSE). 

  • Tuesday April 12, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Six-vertex model and the KPZ universality class

    Amol Aggarwal, Columbia University

    In this talk, we explain recent results relating the six-vertex model and the Kardar-Parisi-Zhang (KPZ) universality class. In particular, we describe how the six-vertex model can be used to analyze stochastic interacting particle systems, such as asymmetric exclusion processes, and how infinite-volume pure states of the ferroelectric six-vertex model exhibit fluctuations of order $N^{1/3}$, a characteristic feature of systems in the KPZ universality class. 
     

  • Tuesday April 19, 2022 at 15:30, Temple (Wachman Hall 617)

    Understanding the upper tail behaviour of the KPZ equation via the tangent method

    Milind Hegde, Columbia University

    The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, closely connected to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.

  • Tuesday April 26, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Multiplicative chaos of the Brownian loop soup

    Antoine Jego, MSRI

    On the one hand, the 2D Gaussian free field (GFF) is a log-correlated Gaussian field whose exponential defines a random measure: the multiplicative chaos associated to the GFF, often called Liouville measure. On the other hand, the Brownian loop soup is an infinite collection of loops distributed according to a Poisson point process of intensity $\theta$ times a loop measure. At criticality ($\theta = 1/2$), its occupation field is distributed like half of the GFF squared (Le Jan's isomorphism). The purpose of this talk is to understand the infinitesimal contribution of one loop to Liouville measure in the above coupling. This work is not restricted to the critical intensity and provides the natural notion of multiplicative chaos associated to the Brownian loop soup when $\theta$ is not equal to $1/2$. 
     

  • Tuesday September 6, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Fractal Geometry of the KPZ equation

    Promit Ghosal, MIT
      
    The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to the KPZ universality class. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention a work in progress with Jaeyun Yi for the (2+1)-d case. 
     

  • Tuesday September 13, 2022 at 15:30, Temple (Wachman Hall 617)

    Ballistic annihilation

    Matthew Junge, Baruch College
     
    In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.

  • Tuesday September 20, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    A central limit theorem for square ice

    Wei Wu, NYU Shanghai

    In the area of statistical mechanics, an important open question is to show that the height function associated with the square ice model (i.e., planar six vertex model with uniform weights), or equivalently with uniform graph homeomorphisms, converges to a continuum Gaussian free field in the scaling limit. I will review some recent results about this model, including that the single point height function, upon renormalization, converges to a Gaussian random variable.

  • Tuesday September 27, 2022 at 15:30, Temple (Wachman Hall 617)

    On roots of random trigonometric polynomials and related models

    Hoi Nguyen, Ohio State University
     
    In this talk, we will discuss various basic statistics of the number of real roots of random trigonometric polynomials, as well as the minimum modulus and the nearest roots statistics to the unit circle of Kac polynomials. We will emphasize the universality aspects of all these problems.
     
    Based on joint works with Cook, Do, O. Nguyen, Yakir and Zeitouni.

  • Tuesday October 4, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Yaglom-type limits for branching Brownian motion with absorption in the slightly subcritical regime

    Jiaqi Liu, Penn

    Branching Brownian motion is a random particle system which incorporates both the tree-like structure and the diffusion process. In this talk, we consider a slightly subcritical branching Brownian motion with absorption, where particles move as Brownian motion with drift, undergo dyadic fission at a constant rate, and are killed upon hitting the origin. We are interested in the asymptotic behaviors of the process conditioned on survival up to a large time t as the process approaches criticality. Results like this are called Yaglom-type results. Specifically, we will talk about the construction of the Yaglom limit law, Yaglom-type limits for the number of particles and the maximal displacement. Based on joint work with Julien Berestycki, Bastien Mallein and Jason Schweinsberg. 
     

  • Tuesday October 11, 2022 at 15:30, Temple (Wachman Hall 617)

    Ergodicity and synchronization of the KPZ equation

    Chris Janjigian, Purdue University
     
    The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to have stationary distributions modulo additive constants given by Brownian motion with drift. In this talk, we will discuss some results-in-progress which show that these distributions are totally ergodic and present some progress toward the conjecture that these are the only ergodic stationary distributions of the KPZ equation. The talk will discuss our coupling of Hopf-Cole solutions, which enables us to study the KPZ equation started from any measurable function valued initial condition. Through this coupling, we give a sharp characterization of when such solutions explode, show that all non-explosive functions become instantaneously continuous, and then study the problem of ergodicity on a natural topology on the space of non-explosive continuous functions (mod constants) in which the equation defines a Feller process. We show that any ergodic stationary distribution on this space is either a Brownian motion with drift or a process of a very peculiar form which will be described in the talk. 

    Based on joint works with Tom Alberts, Firas Rassoul-Agha, and Timo Seppäläinen.

  • Tuesday October 18, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Graph limits and graph homomorphism density inequalities

    Fan Wei, Princeton University

    Graph limits is a recently developed powerful theory in studying large (weighted) graphs from a continuous and analytical perspective. It is particularly useful when studying subgraph homomorphism density, which is closely related to graph property testing, graph parameter estimation, and many central questions in extremal combinatorics. In this talk, we will show how the perspective of graph limits helps with graph homomorphism inequalities and how to make advances in a common theme in extremal combinatorics: when is the random construction close to optimal? We will also show some hardness results for proving general theorems in graph homomorphism density inequalities. 
     

  • Tuesday October 25, 2022 at 15:30, Temple (Wachman Hall 617)

    Optimal delocalization for generalized Wigner matrices

    Lucas Benigni, Université de Montréal

    We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.

  • Tuesday November 1, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Sandpiles

    Ahmed Bou-Rabee, Cornell University

    I will introduce the Abelian sandpile model and discuss its large-scale behavior in random environments and on different lattices. There are many open questions. 
     

  • Tuesday November 8, 2022 at 15:30, Temple (Wachman Hall 617)

    Convergence of densities of spatial averages of the stochastic heat equation

    Şefika Kuzgun, University of Rochester
     
    Let $u$ be the solution to the one-dimensional stochastic heat equation driven by a space-time white noise with constant initial condition. The purpose of this talk is to present a recent result on the uniform convergence of the density of the normalized spatial averages of the solution $u$ on an interval $[-R,R]$, as $R$ tends to infinity, to the density of the standard normal distribution, assuming some non-degeneracy and regularity conditions on the diffusion coefficient. These results are based on the combination of Stein's method for normal approximations and Malliavin calculus techniques. This is a joint work with David Nualart.

  • Tuesday November 29, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

    (at UPenn) Non-backtracking spectra of random hypergraphs and community detection

    Yizhe Zhu, UC Irvine
     
    The stochastic block model has been one of the most fruitful research topics in community detection and clustering. Recently, community detection on hypergraphs has become an important topic in higher-order network analysis. We consider the detection problem in a sparse random tensor model called the hypergraph stochastic block model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al (2015). We characterize the spectrum of the non-backtracking operator for sparse random hypergraphs and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, the community detection problem can be reduced to an eigenvector problem of a non-normal matrix constructed from the adjacency matrix of the hypergraph. Based on joint work with Ludovic Stephan (EPFL). 
     

Probability Seminars 2023

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

 

  • Tuesday January 24, 2023 at 15:30, Temple (Wachman Hall 617)

    Rare transitions in noisy heteroclinic networks

    Yuri Bakhtin, Courant Institute

    We study white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude, and we describe the underlying mechanism. A metastability picture emerges, with a hierarchy of time scales and clusters of accessibility, similar to the classical Freidlin-Wentzell picture but with shorter transition times. We discuss applications of our results to homogenization problems and to the invariant distribution asymptotics. At the core of our results are local limit theorems for exit distributions obtained via methods of Malliavin calculus. Joint work with Hong-Bin Chen and Zsolt Pajor-Gyulai.

  • Tuesday January 31, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

    Infinite cycles in the interchange process in five dimensions

    Dor Elboim, Princeton University 
     
    In the interchange process on a graph $G=(V, E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $\pi _\beta: V\to V$ is formed for any time $\beta >0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. 
     
    We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $\beta$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions. 
     
    In our proof, we study a self-interacting random walk called the cyclic time random walk. Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.
     
    This is a joint work with Allan Sly.

  • Tuesday February 7, 2023 at 15:30, Temple (Wachman Hall 617)

    Mean-field games: asymptotics and refined convergence results

    Kavita Ramanan, Brown University 
     
    A mean-field game is a game with a continuum of players,  describing the limit as n tends to infinity of Nash equilibria of certain n-player games, in which agents interact symmetrically through the empirical measure of their state processes. We first study the asymptotic behavior of Nash equilibria in static games with a large number of agents. In particular, we establish law of large number limits and large deviation principles for the set of Nash equilibria and discuss applications to congestion games and the price of anarchy. Then we discuss stochastic differential games, which are often understood via the so-called "master equation", which is an infinite-dimensional PDE for the value function. We will show how analysis of sufficiently smooth solutions to the master equation play a role in analyzing large deviation principles for mean-field games. This is based on joint works with Francois Delarue and Daniel Lacker.

  • Tuesday February 14, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

    KPZ on a large torus

    Yu Gu, University of Maryland
     
    I will present the recent work with Tomasz Komorowski and Alex Dunlap in which we derived optimal variance bounds on the solution to the KPZ equation on a large torus, in certain regimes where the size of the torus increases with time. We only use stochastic calculus and I will try to give a heuristic explanation of the 2/3 and 1/3 exponents in the 1+1 KPZ universality class.

  • Tuesday February 21, 2023 at 15:30, Temple (Wachman Hall 617)

    The nonlinear stochastic heat equation in the critical dimension

    Alex Dunlap, Courant Institute

    I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and earlier joint work with Yu Gu.

  • Tuesday February 28, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

    Functional inequalities on the space of d-regular directed graphs, with applications to mixing

    Konstantin Tikhomirov, Carnegie Mellon University

    We consider the space of d-regular directed simple graphs, where two graphs are connected whenever there is a simple switching operation transforming one graph to the other. For constant d, we prove optimal bounds on the modified Log-Sobolev constant of the associated Markov chain on the space of graphs. This implies that the total variation mixing time of the chain is of order n log(n), which settles an old open problem. Based on joint work with Pierre Youssef.

  • Tuesday March 14, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

    Liouville conformal field theory and the quantum zipper

    Morris Ang, Columbia University
     
    Sheffield showed that conformally welding a \gamma-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with parameter \kappa = \gamma^2 as the interface, and Duplantier-Miller-Sheffield proved similar stories for \kappa = 16/\gamma^2 for \gamma-LQG surfaces with boundaries decorated by looptrees of disks or by continuum random trees. We study these dynamics for LQG surfaces coming from Liouville conformal field theory (LCFT). At stopping times depending only on the curve, we give an explicit description of the surface and curve in terms of LCFT and SLE. This has applications to both LCFT and SLE. We prove the boundary BPZ equation for LCFT, which is crucial to solving boundary LCFT. With Yu we prove the reversibility of whole-plane SLE for \kappa ≥ 8 via a novel radial mating-of-trees.

  • Tuesday March 21, 2023 at 15:30, Temple (Wachman Hall 617)

    Algorithmic barriers from intricate geometry in random computational problems

    Eren C. Kızıldağ, Columbia University
     
    Many computational problems involving randomness exhibit a statistical-to-computational gap (SCG): the best known polynomial-time algorithm performs strictly worse than the existential guarantee. In this talk, we focus on the SCG of the symmetric binary perceptron (SBP), a random constraint satisfaction problem as well as a toy model of a single-layer neural network. We establish that the solution space of the SBP exhibits intricate geometrical features, known as the multi Overlap Gap Property (m-OGP). By leveraging the m-OGP, we obtain nearly sharp hardness guarantees against the class of stable and online algorithms, which capture the best known algorithms for the SBP. Our results mark the first instance of intricate geometry yielding tight algorithmic hardness against classes beyond stable algorithms.

    Time permitting, I will discuss how the same program extends also to other models, including (a) discrepancy minimization, and (b) random number partitioning problem. 

    Based on joint works with David Gamarnik, Will Perkins, and Changji Xu.

  • Tuesday March 28, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

    Phase transition in mean-field models

    Wenpin Tang, Columbia University
     
    In this talk, I will discuss two mean-field models in which a certain phase transition occurs. I first describe McKean-Vlasov equations involving hitting times which arise as the mean-field limit of particle systems with annihilation. One such example is the super-cool Stefan problem. It is well known that such a system may have blow-ups. We provide some sufficient conditions on the model data to assure either blow-ups or no blow-ups. In the second part, I will discuss the convergence rate of second-order mean-field games to first-order ones, motivated from numerical challenges in first-order mean-field PDEs and the weak noise theory in KPZ universality. When the Hamiltonian and the coupling function have a certain growth, the rate is independent of the dimension; on the other hand, the rate decays in dimension (curse of dimensionality) when the Hamiltonian and the coupling function have small growth. These are based on joint work with Yuming Paul Zhang.

  • Tuesday April 4, 2023 at 15:30, Temple (Wachman Hall 617)

    A Berry-Esseen theorem and Edgeworth expansions for inhomogeneous elliptic Markov chains

    Yeor Hafouta, University of Maryland
     
    We obtain optimal rates in the central limit theorem (CLT) for additive functionals of uniformly elliptic inhomogeneous Markov chains without any assumptions on the growth rates of the variance of the underlying partial sums. (The CLT itself is due to Dobrushin (1956) and it holds in greater generality.)

    We will also discuss Edgeworth expansions (i.e., the correction terms in the CLT) of order one for general classes of functionals, which provide a structural characterization of having better than optimal CLT rates.

    Finally, for several classes of additive functionals (e.g., Holder continuous), we will provide optimal conditions for Edgeworth expansions of an arbitrary order.

    The talk is based on a joint work with Dmitry Dolgopyat.

  • Tuesday April 11, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

    Ising model on locally tree-like graphs: uniqueness of solutions to cavity equations

    Qian Yu, Princeton University

    In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there is at most one non-trivial fixed point for Ising models with zero or certain random external fields. Previously this was only known for sufficiently ``low-temperature'' models. 

    Our result simultaneously closes the following 6 conjectures in the literature: 1) independence of robust reconstruction accuracy to leaf noise in broadcasting on trees (Mossel-Neeman-Sly'16); 2) uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM (Kanade-Mossel-Schramm'16); 3) optimality of local algorithms for 2-SBM under noisy side information (Mossel-Xu'16); 4) uniqueness of BP fixed point in broadcasting on trees in the Gaussian (large degree) limit (ibid); 5) boundary irrelevance in broadcasting on trees (Abbe-Cornacchia-Gu-Polyanskiy'21); 6) characterization of entropy (and mutual information) of community labels given the graph in 2-SBM (ibid). 

    This is a joint work with Yury Polyanskiy.

  • Tuesday April 11, 2023 at 16:30, Penn (David Rittenhouse Lab 4C4)

    Cutoff profile of the colored ASEP: GOE Tracy-Widom

    Lingfu Zhang, UC Berkeley

    In this talk, I will discuss the colored Asymmetric Simple Exclusion Process (ASEP) in a finite interval. This Markov chain is also known as the biased card shuffling or random Metropolis scan, and its study dates back to Diaconis-Ram (2000). A total-variation cutoff was proved for this chain a few years ago using hydrodynamic techniques (Labbé-Lacoin, 2016). In this talk, I will explain how to obtain more precise information on its cutoff, specifically to establish the conjectured GOE Tracy-Widom cutoff profile. The proof relies on coupling arguments, as well as symmetries obtained from the Hecke algebra. I will also discuss some related open problems.

  • Tuesday April 18, 2023 at 15:30, Temple (Wachman Hall 617)

    Minimal surfaces in random environment

    Ron Peled, Tel Aviv University, IAS and Princeton University

    A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of d-dimensional minimal surfaces in a (d+n)-dimensional random environment. Specializing to a model that we term harmonic MSRE, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as a scaling relation that ties together these two types of fluctuations.

    Joint work with Barbara Dembin, Dor Elboim and Daniel Hadas.

  • Tuesday April 25, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

    Stochastic waves on metric graphs and their genealogies

    Louis Fan, Indiana University
     
    Stochastic reaction-diffusion equations are important models in mathematics and in applied sciences such as spatial population genetics and ecology. These equations describe a quantity (density/concentration of an entity) that evolves over space and time, taking into account random fluctuations. However, for many reaction terms and noises, the solution notion of these equations is still missing in dimension two or above, hindering the study of the spatial effect on stochastic dynamics through these equations.

    In this talk, I will discuss a new approach, namely, to study these equations on general metric graphs that flexibly parametrize the underlying space. This enables us to not only bypass the ill-posedness issue of these equations in higher dimensions, but also assess the impact of space and stochasticity on the coexistence and the genealogies of interacting populations. We will focus on the computation of the probability of extinction, the quasi-stationary distribution, the asymptotic speed and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type.

  • Tuesday September 5, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Stochastic quantization of Yang-Mills in 2D and 3D

    Hao Shen, University of Wisconsin-Madison
     
    Quantum Yang-Mills model is a type of quantum field theory with gauge symmetry. The rigorous construction of quantum Yang-Mills is a central problem in mathematical physics. Stochastic quantization formulates the problem as stochastic dynamics, which can be studied using tools from analysis, PDE and stochastic PDE. We will discuss stochastic quantization of Yang-Mills on the 2 and 3 dimensional tori. To this end we need to address a number of questions, such as the construction of a singular orbit space, together with class gauge invariant observables (singular holonomies or Wilson loops), solving a stochastic PDE using regularity structures, and projecting the solution to the orbit space. Mostly based on joint work with Chandra, Chevyrev and Hairer.

  • Tuesday September 12, 2023 at 15:30, Temple (Wachman Hall 617)

    Statistical mechanics of Log and Riesz interactions

    Luke Peilen, Temple University

    We study the statistical mechanics of the log gas, an interacting particle system with applications in random matrix theory and statistical physics, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on a novel next order energy quantity that are valid down to microscopic length scales. Simultaneously, we exhibit a control on fluctuations of linear statistics that is also valid down to microscopic scales. Using these local laws, we exhibit for the first time a CLT at arbitrary mesoscales, improving upon previous results of Bekerman and Lodhia.
     
    The methods we use are suitable for generalization to higher dimensional Riesz interactions; we will discuss some generalizations of the above approach and partial results for the Riesz gas in higher dimensions.

  • Tuesday September 19, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Random walks in (Dirichlet) random environments with jumps on $\mathbb{Z}$

    Daniel Slonim, University of Virginia
     
    We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.

  • Tuesday September 26, 2023 at 15:30, Temple (Wachman Hall 617)

    Large deviations of the KPZ equation and most probable shapes

    Yier Lin, University of Chicago

    The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments.

    This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.

  • Tuesday October 3, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Geometry of the doubly-periodic Aztec dimer model

    Tomas Berggren, MIT
     
    Random dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. In this talk, we will discuss the doubly-periodic Aztec diamond dimer model of growing size, with arbitrary periodicity and only mild conditions on the edge weights. In this limit, we see three types of macroscopic regions — known as rough, smooth and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of these regions, can be described in terms of an associated amoeba and an action function. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. We will also discuss the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures. Joint work with Alexei Borodin.

  • Tuesday October 10, 2023 at 15:30, Temple (Wachman Hall 617)

    A rigorous approximation of a certain random Fermi-Pasta-Ulam-Tsingou (FPUT) lattice by the Korteweg-De Vries (KdV) equation

    Joshua McGinnis, UPenn

    We review recent results regarding the rigorous approximation of 1D and 2D disordered (random, independent masses and/or springs) harmonic lattices by effective wave equations in the long wave limit. In this linear setting, we show the homogenization argument and highlight the tools used from probability theory to control the stochastic error terms such as the Law of the Iterated Logarithm and Hoeffding’s inequality. With our discussion of the linear problem serving as a springboard, we then present a new result regarding the approximation of an FPUT lattice with random masses by a KdV equation. Specifically, we are able to bound the approximation error in terms of the small parameter from the long wave scaling in an almost sure sense. In our theorem, we require a technical condition on the random masses, which we call transparency. Our proof relies on the incorporation of an auto-regressive process into an approximating ansatz, which itself is approximated by solutions to the KdV equation. We discuss the role of the auto-regressive process as well as the condition of transparency in the proof and give numerical evidence supporting the result. We conclude by discussing open questions such as the apparent lack of KdV dynamics in an FPUT lattice with independent, random masses.

  • Tuesday October 17, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Sample duality

    Adrián González-Casanova, UC Berkeley

    Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x". Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique is useful, including an application to the Simple Exclusion Process with reservoirs. The last part of the lecture is based in recent joint work with Simone Floriani.

  • Tuesday October 24, 2023 at 15:30, Temple (Wachman Hall 617)

    An iterative approach to estimating integrated volatility

    Cooper Boniece, Drexel University

    The quadratic variation of a semimartingale plays an important role in a variety of applications, particularly so in financial econometrics, where it is closely linked to volatility.  It contains information pertaining to both continuous and discontinuous path behavior of the underlying process, and separating its continuous and discontinuous parts based on high-frequency observations is a problem that has been tackled through a variety of approaches to-date.
     
    However, despite the favorable asymptotic statistical properties of many of these approaches, their use in practice requires heuristic selection of tuning parameters that can greatly impact their estimation performance.
      
    In this talk, I will discuss some recent work concerning an iterative approach that circumvents the "tuning problem."

    This is based on joint work with J. E. Figueroa-López and Y. Han.

  • Tuesday October 31, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Fundamental limits of low-rank matrix estimation: information-theoretic and computational perspectives

    Yuchen Wu, Penn
     
    Many statistical estimation problems can be reduced to the reconstruction of a low-rank n×d matrix when observed through a noisy channel. While tremendous positive results have been established, relatively few works focus on understanding the fundamental limitations of the proposed models and algorithms. Understanding such limitations not only provides practitioners with guidance on algorithm selection, but also spurs the development of cutting-edge methodologies. In this talk, I will present some recent progress in this direction from two perspectives in the context of low-rank matrix estimation. From an information-theoretic perspective, I will give an exact characterization of the limiting minimum estimation error. Our results apply to the high-dimensional regime n,d→∞ and d/n→∞ (or d/n→0) and generalize earlier works that focus on the proportional asymptotics n,d→∞, d/n→δ∈(0,∞). From an algorithmic perspective, large-dimensional matrices are often processed by iterative algorithms like power iteration and gradient descent, thus encouraging the pursuit of understanding the fundamental limits of these approaches. We introduce a class of general first order methods (GFOM), which is broad enough to include the aforementioned algorithms and many others. I will describe the asymptotic behavior of any GFOM, and provide a sharp characterization of the optimal error achieved by the GFOM class.

  • Tuesday November 7, 2023 at 15:30, Temple (Wachman Hall 617)

    Relative instability and concentration of equilibria in non-gradient dynamics

    Pax Kivimae, Courant Institute, NYU

    A classical picture in the theory of complex high-dimensional random functions is that an exponentially large number of critical points causes the gradient dynamics of the function to become slow and "glassy", becoming trapped in local minima. In non-gradient dynamics however, another case is possible. Here, one may have an exponentially large number of equilibria, but have none that are stable, leading to an endless cycle of wandering around saddles. This is believed to occur when the strength of the non-gradient terms is brought past a certain point, a phenomenon coined by Ben Arous, Fyodorov, and Khoruzhenko as the relative-absolute instability transition, and since predicted to occur in a variety of models.

    We confirm such a transition occurs in the case of the asymmetric p-spin model, the first such rigorous confirmation of the existence of this transition in any model. To do so, we demonstrate concentration of the quenched complexity of stable and general equilibria around their annealed values. Our methods rely on generalizing the recent framework of Ben Arous, Bourgade, and McKenna on the Kac-Rice formula to the non-relaxational case, as well as a computation of moments of the characteristic polynomial of the elliptic ensemble.

  • Tuesday November 14, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Spectral gap estimates for mixed $p$-spin models at high temperature

    Arka Adhikari, Stanford University

    We consider general mixed $p$-spin mean-field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the $N$-spin system to that of suitably conditioned subsystems.

    Based on joint work w/ C. Brennecke, C. Xu, and H.-T. Yau.

  • Tuesday November 28, 2023 at 15:30, Temple (Wachman Hall 617)

    KPZ fluctuations in the planar stochastic heat equation

    Alejandro Ramírez, NYU Shanghai
     
    We consider Wick-ordered solutions to the planar stochastic heat equation, corresponding to a Skorokhod interpretation in the Duhamel integral representation of the equation. We prove that the fluctuations far from the center are given by the stochastic heat equation. This talk is based on a joint work with Jeremy Quastel and Balint Virag.

  • Tuesday December 5, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Stationary measures for integrable polymers on a strip

    Zongrui Yang, Columbia University 

    We prove that the stationary measures for the geometric last passage percolation and log-gamma polymer models on a diagonal strip are given by the marginals of objects we call two-layer Gibbs measures. Taking an intermediate disorder limit of the log-gamma polymer stationary measure, we recover the conjectural description of the open KPZ equation stationary measure for all choices of boundary parameters. This is a joint work with Guillaume Barraquand and Ivan Corwin.

Probability Seminars 2024

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

 

  • Tuesday January 30, 2024 at 15:30, Wachman 617

    Log-concavity in 1-d Coulomb gas ensembles 

    Mokshay Madiman, University of Delaware

    The ordered elements in several one-dimensional Coulomb gas ensembles arising in probability and mathematical physics are shown to have log-concave distributions. Examples include the beta ensembles with convex potentials (in the continuous setting) and the orthogonal polynomial ensembles (in the discrete setting). In particular, we prove the log-concavity of the Tracy-Widom β distributions, Airy distribution, and Airy-2 process. Log-concavity of last passage times in percolation is proven using their connection to Meixner ensembles. We then obtain the log-concavity of top rows of Young diagrams under Poissonized Plancherel measure, which is the Poissonized version of a conjecture of Chen. This is ongoing joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.

  • Tuesday February 6, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Asymptotic topological statistics of Gaussian random zero sets

    Zhengjiang Lin, Courant Institute, NYU
     
    We will briefly discuss some asymptotic topological statistics of Gaussian random zero sets, which include a random distribution on knots as a special case. We will also discuss some results on zero sets of random Laplacian eigenfunctions, which are related to Courant’s nodal domain theorem and Milnor-Thom’s theorem on Betti numbers of real algebraic varieties.

  • Tuesday February 13, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

    The spherical mixed p-spin glass at zero temperature

    Yuxin Zhou, University of Chicago

    In this talk, I will discuss the spherical mixed p-spin glass model at zero temperature. I will present some recent results that classify the possible structure of the functional ordered parameter. For spherical p+s spin glasses, we classify all possibilities for the Parisi measure as a function of the model. Moreover, for the spherical spin models with n components, the Parisi measure at zero temperature is at most n-RSB or n-FRSB. Some of these results are jointly with Antonio Auffinger.

  • Tuesday February 20, 2024 at 15:30, Wachman 617

    A matrix model for conditioned Stochastic Airy

    Brian Rider, Temple University

    There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure. What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. I will describe a general beta matrix model for which the appropriate double scaling limit is the Stochastic Airy Operator, conditioned on having no eigenvalues below a fixed level.  I know of no other random matrix double scaling fully characterized outside of beta = 2. This is work in progress with J. Ramirez (University of Costa Rica).

  • Tuesday February 27, 2024 at 15:30, Wachman 617

    Periodic orbits of stochastic Hamiltonian ODEs

    Fraydoun Rezakhanlou, University of California, Berkeley

    According to Conley-Zehnder's theorem, any periodic Hamiltonian ODE in $\mathbb{R}^{2n}$ has at least $2n+1$ geometrically distinct periodic orbits. For a stochastically stationary Hamiltonian ODE, the set of periodic orbits yields a translation invariant random process. In this talk, I will discuss an ergodic theorem for the density of periodic orbits, and formulate some open questions which are the stochastic variants of Conley-Zehnder's theorem.

  • Tuesday March 12, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Weak universality in random walks in random environments

    Sayan Das, University of Chicago

    We consider one-dimensional simple random walks whose all one-step transition probabilities are iid [0,1]-valued mean 1/2 random variables. In this talk, we will explain how under a certain moderate deviation scaling the quenched density of the walk converges weakly to the Stochastic Heat Equation with multiplicative noise. Our result captures universality in the sense that it holds for all non-trivial laws for random environments. Time permitting, we will discuss briefly how our proof techniques depart from the existing techniques in the literature. Based on a joint work with Hindy Drillick and Shalin Parekh.

  • Tuesday March 26, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Contact process on large networks

    Oanh Nguyen, Brown University
     
    The contact process serves as a model for the spread of epidemics on networks, with three popular variations: the Susceptible-Infected-Recovered-Susceptible (SIRS), SIR, and SIS. Our focus lies in understanding the temporal evolution of these processes, especially regarding survival time and its associated phase transitions. I will provide a brief overview of related literature, recent progress, and open problems.

  • Tuesday April 2, 2024 at 15:30, Wachman 617

    A generalization of hierarchical exchangeability on trees to Directed Acyclic Graphs

    Paul Jung, Fordham University

    We discuss a class of partially exchangeable random arrays which generalizes the notion of hierarchical exchangeability introduced in Austin and Panchenko (2014). We say that our partially exchangeable arrays are DAG-exchangeable since their partially exchangeable structure is governed by a collection of Directed Acyclic Graphs (DAG). More specifically, such a random array is indexed by N^|V| for some DAG, G = (V,E), and its exchangeability structure is governed by the edge set E. We prove a representation theorem which generalizes the Aldous-Hoover and Austin-Panchenko representation theorems.

  • Tuesday April 9, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Scaling limits in dimers and tableaux

    Zhongyang Li, University of Connecticut

    We investigate limit shapes and height fluctuations in statistical mechanical models, such as dimers and lecture hall tableaux, through the asymptotics of symmetric polynomials. Confirming a conjecture by Corteel, Keating, and Nicoletti, we show that the rescaled height functions' slopes in the scaling limit of lecture hall tableaux adhere to a complex Burgers equation.

  • Tuesday April 16, 2024 at 15:30, Wachman 617

    The shape of the front of multidimensional branching Brownian motion

    Yujin Kim, Courant Institute, NYU

    The extremal process of branching Brownian motion (BBM) —i.e., the collection of particles furthest from the origin— has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point— the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.

  • Tuesday April 23, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

    Programmable matter and emergent computation

    Dana Randall, Georgia Tech

    Programmable matter explores how collections of computationally limited agents acting locally and asynchronously can achieve some useful coordinated behavior.  We take a stochastic approach using techniques from randomized algorithms and statistical physics to develop distributed algorithms for emergent collective behaviors that give guarantees and are robust to failures.

  • Tuesday April 30, 2024 at 15:30, Wachman 617

    Homogenization of nonconvex Hamilton-Jacobi equations in stationary ergodic media

    Atilla Yilmaz, Temple University

    I will start with a self-contained introduction to the homogenization of inviscid (first-order) and viscous (second-order) Hamilton-Jacobi (HJ) equations in stationary ergodic media in any dimension. After a brief account of the now-classical works that are concerned with periodic media or convex Hamiltonians, I will return to the general setting and outline the results obtained in the last decade that: (i) established homogenization for inviscid HJ equations in one dimension; and (ii) provided counterexamples to homogenization in the inviscid and viscous cases in dimensions two and higher. Finally, I will present my recent joint work with E. Kosygina in which we prove homogenization for viscous HJ equations in one dimension, and also describe how the solution of this problem qualitatively differs from that of its inviscid counterpart.

  • Tuesday September 3, 2024 at 15:30, Wachman 617

    Vector-valued concentration on the symmetric group

    Mira Gordin, Princeton University
     
    Concentration inequalities for real-valued functions are well understood in many settings and are classical probabilistic tools in theory and applications -- however, much less is known about concentration phenomena for vector-valued functions. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group. Furthermore, we discuss the implications of this result regarding the distortion of embeddings of the symmetric group into Banach spaces, a question which is of interest in metric geometry and algorithmic applications. We build on prior work of Ivanisvili, van Handel, and Volberg (2020) who proved a vector-valued inequality on the discrete hypercube, resolving a conjecture of Enflo in the metric theory of Banach spaces. This talk is based on joint work with Ramon van Handel.

  • Tuesday September 10, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

    Functional limit theorems for local functionals of dynamic point processes

    Efe Onaran, UPenn
     
    I will present functional limit theorems for local, additive, interaction functions of temporally evolving point processes. The dynamics are those of a spatial Poisson process on the flat torus with points subject to a birth-death mechanism, and which move according to Brownian motion while alive. The results reveal the existence of a phase diagram describing at least three distinct structures for the limiting processes, depending on the extent of the local interactions and the speed of the Brownian movements. The proofs, which identify three different limits, rely heavily on Malliavin-Stein bounds on a representation of the dynamic point process via a distributionally equivalent marked point process. Based on a joint work with Omer Bobrowski and Robert J. Adler.

  • Tuesday September 17, 2024 at 15:30, Wachman 617

    An involution framework for Metropolis-Hastings algorithms on general state spaces

    Cecilia Mondaini, Drexel University
     
    Metropolis-Hastings algorithms are a common type of Markov Chain Monte Carlo method for sampling from a desired probability distribution. In this talk, I will present a general framework for such algorithms which is based on a fundamental involution structure on a general state space, and encompasses several popular algorithms as special cases, both in the finite- and infinite-dimensional settings. In particular, these include random walk, preconditioned Crank-Nicolson (pCN), schemes based on a suitable Langevin dynamics such as the Metropolis Adjusted Langevin algorithm (MALA), and also ones based on Hamiltonian dynamics including several variants of the Hamiltonian Monte Carlo (HMC) algorithm. In fact, with a slight generalization of our first framework, we are also able to cover algorithms that generate multiple proposals at each iteration. These have the potential of providing efficient sampling schemes through the use of modern parallel computing resources. Here we derive several generalizations of the aforementioned algorithms following as special cases of this multiproposal framework. To illustrate the effectiveness of these sampling procedures, we present applications in the context of some Bayesian inverse problems in fluid dynamics. This is based on joint works with N. Glatt-Holtz (Indiana University), A. Holbrook (UCLA), and J. Krometis (Virginia Tech).

  • Tuesday September 24, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

    Coefficientwise Hankel-total positivity in enumerative combinatorics 

    Alan Sokal, University College London
     
    The abstract (which involves quite a bit of TeX) is available here.

  • Tuesday October 8, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

    KPZ equation from driven lattice gases

    Kevin Yang, Harvard University 

    We will discuss a family of exclusion processes in one spatial dimension, where the random walk particles feel a drift whose speed depends on the local particle configuration. We show that their height fluctuations have a large-N limit given by the KPZ equation with an additional linear transport term. To our knowledge, it is the first KPZ result for a class of particle systems where the invariant measures are not explicit or known. This result also extends a prior series of works on deriving KPZ from stochastic Hamilton-Jacobi equations of Hairer, Quastel, Shen, Xu, and others.

  • Tuesday October 22, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

    Reconstruction on hypertrees

    Yuzhou Gu, NYU

    We develop new methods for analyzing information propagation in branching processes. Our approach is applied to the broadcasting on hypertrees (BOHT) problem, where we obtain the exact reconstruction threshold for a wide range of parameters. As a consequence, we establish the weak recovery threshold for the hypergraph stochastic block model and the condensation threshold for the random NAE-SAT problem in the corresponding parameter regimes, resolving conjectures made by physicists. Our method introduces a rigorous version of population dynamics and improves robust reconstruction analysis. The core of our analysis relies on information-theoretic methods for channel comparison.

  • Tuesday October 29, 2024 at 15:30, Wachman 617

    The diffusion limit of the Aldous chain on the space of continuum trees

    Douglas Rizzolo, University of Delaware

    Tree-valued dynamics arise in applications in many areas such as computer science, machine learning, and phylogenetics, often in the context of Markov chain Monte Carlo inference. The immense size of phylogenetic trees has motivated a growing literature on asymptotic properties of such Markov chains and their scaling limits and continuum analogs.  In the 1990's David Aldous conjectured the existence of a scaling limit for a tree-valued Markov chain that can be thought of as the simple random walk on binary trees.  Despite significant interest, constructing the limiting process using traditional methods such as generators or martingale problems is challenging and, indeed, remains open.  In this talk we will discuss the recent resolution of Aldous's conjecture using a novel pathwise construction.  Along the way, we will discuss how some important intermediate processes we construct are related to integrable probability, specifically to up-down chains on branching graphs.

  • Tuesday November 12, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

    Limits of large contingency tables

    Sumit Mukherjee, Columbia University

    We explore the limiting structure of a large contingency table (in cut metric), when the row and column marginals converge in a suitable sense. Our results go beyond the classical contingency table framework, and allows for general matrices with real entries chosen from an arbitrary base measure.

  • Tuesday November 19, 2024 at 15:30, Wachman 617

    The dipole phase transition in the 2D Coulomb gas

    Jeanne Boursier, Columbia University
     
    The 2D two-component Coulomb gas is expected to exhibit an infinite sequence of phase transitions driven by the divergence of 2k-poles, accumulating at the Berezinskii-Kosterlitz-Thouless temperature. I will discuss joint work with Sylvia Serfaty in which we provide a rigorous proof of this "dipole transition." Our proof is based on the analysis of dipole pairs and large deviations techniques.

  • Tuesday December 3, 2024 at 15:30, Wachman 617

    The No-U-Turn sampler: reversibility and mixing time

    Nawaf Bou-Rabee, Rutgers University

    The No-U-Turn sampler (NUTS) is arguably one of the most important Markov chain Monte Carlo methods out there, but its recursive architecture has long made it challenging to understand.  This talk will provide a clearer picture of how NUTS operates and why it performs well in high-dimensional problems.  Specifically, I will present a concise proof of NUTS’ reversibility by interpreting it as an auxiliary variable method, in the sense of Section 2.1 of arxiv.org/abs/2404.15253.  This novel auxiliary variable approach offers significant flexibility for constructing transition kernels that are reversible with respect to a given target distribution, and includes as special cases Metropolis-Hastings, slice samplers, proximal samplers, and existing locally adaptive HMC methods.  Next, I will present the first mixing time guarantee for NUTS, based on couplings and concentration of measure, which aligns well with empirical observations. Specifically, the mixing time of NUTS, when initialized in the concentration region of the canonical Gaussian measure, scales as d^{1/4}, up to log factors, where d is the dimension (see arxiv.org/abs/2410.06978). This scaling is expected to be sharp (see arxiv.org/abs/1001.4460). A key insight in our analysis is that concentration of measure leads to uniformity in NUTS’ locally adapted transitions.  We formalize this uniformity by using an extension of a recent coupling framework (see arxiv.org/abs/2308.04634) to a broader class of accept/reject chains. NUTS is then interpreted as one such chain, with the accept chain showing more uniform behavior.  This is joint work with Bob Carpenter (Flatiron), Milo Marsden (Stanford), and Stefan Oberdörster (Bonn).