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).