Probability Seminar

In stochastic homogenization, solutions to a heterogeneous equation converge to the solution to a homogeneous equation provided that the coefficients are stationary, ergodic and satisfy a sufficient ellipticity condition. I will explain why certain coarse-grained ellipticity constants appear naturally in homogenization, show that boundedness of the coarse-grained ellipticity constants implies quenched homogenization of the PDE, and compare this to recent results on the random conductance model and the case of a divergence-free drift.

Consider a connected finite graph in which each vertex carries a real number. At each step, an edge (u, v) is chosen uniformly at random, and the numbers at u and v are replaced by their average. This dynamics, known as the repeated averaging process, appears in many contexts, including thermal equilibration, opinion dynamics, wealth exchange, and quantum circuits. All numbers eventually converge to the global average, and we study the speed of convergence in the L1 distance (which is, for example, the Gini index in wealth distributions). On random d-regular graphs, we show a sharp phase transition in this decay, where the L1 distance drops abruptly to zero with a Gaussian profile. Our techniques are robust, and we expect them to extend to more general dynamics on expander graphs. This is joint work in preparation with Dong Yao.

In this talk, I will present the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. A key focus of the talk will be the surgery method we developed to handle these singularities and establish global asymptotics. I will also discuss applications of this result, including the convergence of the characteristic polynomial of random normal matrices to Gaussian Multiplicative Chaos measure. Based on joint work with Paul Bourgade, Guillaume Dubach, and Lisa Hartung.

Coulomb and Riesz gases are interacting particle systems with a wide range of applications in random matrix theory, approximation theory, convex geometry, and diverse areas of physics. We study the statistical mechanics of general Riesz gases at mesoscopic and microscopic length scales, providing controls on fluctuations of linear statistics down to microscopic length scales and establishing for the first time a CLT for fluctuations of linear statistics for general two-dimensional Riesz gases.

A novel technical difficulty involves the development of a transport method for general Riesz gases, building on work of Leblé and Serfaty for Coulomb gases, to understand the behavior of the partition function under small perturbations of the external potential. Our study involves several questions concerning degenerate, singular elliptic PDE and fractional operators.

This is based on joint work with S. Serfaty.

This is based on a joint work with Joffrey Mathien. We establish new instances of the cutoff phenomenon for geodesic paths and for the Brownian motion on compact hyperbolic manifolds. We prove that for any fixed compact hyperbolic manifold, the geodesic path started on a spatially localized initial condition exhibits cutoff. Our work also extends results obtained by Golubev and Kamber on hyperbolic surfaces of large volume to any dimension. More generally, we will discuss ongoing works on the cutoff phenomenon in mixing dynamical systems.

This talk will be devoted to the discrete long-range Gaussian chain with $1/r^\alpha$ interactions. I will introduce the model, its history and phase diagram. In this direction, a first notable result is the existence of a roughening phase transition for $\alpha = 2$ established by Kjaer-Hilhorst and Fröhlich-Zegarlinski. For $\alpha > 2$, the model is not expected to undergo a phase transition and a few important results have been recently obtained: Garban characterised the fluctuations of the chain at high temperature (and, in fact, fully identified its scaling limit) and Coquille–van Enter–Le Ny–Ruszel showed the (qualitative) delocalisation of the chain at every inverse temperature. After discussing these results in more details, I will present some quantitative estimates in the low temperature regime with range exponent $\alpha > 2$ obtained in a joint work with L. Coquille and A. Le Ny.

It is widely observed that overparameterized neural networks—often with more parameters than training samples—can interpolate the training data while still generalizing well. One theoretical approach to this phenomenon studies the dynamics of gradient-based training. Empirically and in several settings theoretically, gradient descent is seen to converge to particular “simpler” solutions among many minimizers, a bias commonly referred to as implicit regularization. Early stopping during the training process can further reduce effective model complexity and often improves generalization.

In this talk, we adopt the mean-field formulation on wide neural networks, representing the network by a probability measure over parameters and viewing training as a gradient flow on Wasserstein space. Building on this viewpoint, we introduce a mean-field control formulation of the training dynamics. This control perspective, together with dynamic programming principle, leads to a mean-field analogue of the Wasserstein-2 distance and provides a framework for analyzing early stopping and implicit regularization.

The dimer model, i.e. random perfect matchings of a bipartite graph, is a classical object about which much is known. As soon as one biases the probability measure by edge weights which are themselves random, very little is known rigorously, though physicists have studied such models for several decades and made extensive predictions. I will discuss a new integrable model in this class (the Gamma-disordered Aztec diamond) which allows us to prove versions of some of these, and also exhibits surprising relations to integrable polymer models in the KPZ universality class which allow one to port results between the two. Joint work with Maurice Duits (KTH), https://arxiv.org/abs/2512.03033.

The chemical distance is the observable that encapsulates the metric structure of percolation clusters. At criticality, heuristics suggest that the chemical distance between two connected points scales quadratically in the extrinsic distance, in line with the analogy to branching random walk. Our work presents an exact statement of this result, where the rescaled two-point chemical distance converges in distribution to a random variable whose density is expressible as a Brownian motion hitting time. The method relies on the robust incipient infinite cluster constructed in our previous work to enforce a decoupling argument that separates neighborhoods of distant pivotal edges. This decoupling tool yields further applications towards studying the mass structure of percolation clusters, i.e. k-point functions, which is necessary in the steps towards a full scaling limit result for the IIC. These projects are joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe. The preprint can be found at https://arxiv.org/abs/2509.06236.

Hambly and Jordan (2004) introduced the series-parallel graph, a random hierarchial lattice that is easy to define: Start with the graph consisting of one edge connecting two terminal nodes.  At each subsequent step of the construction, perform independent coin flips for each edge of the graph, and replace the edge by two edges in series if the coin is heads-up or two edges in parallel if tails.  This results in a sequence of random graphs, which can be interpreted as a resistor network.  Hambly and Jordan showed that the logarithm of the resistance grows linearly if the coins are biased to land more often heads-up.  In this talk, I will discuss what happens in the critical case when fair coins are used.  Starting with a new recursive distributional equation (RDE) proposed by Gurel-Gurevich, I develop a framework for analyzing RDE's based on parabolic PDE theory and use this to characterize the asymptotic behavior of the log. resistance.  In the sub- or supercritical case (where the coins are biased), I discuss a tantalizing connection to the Fisher-KPP equation and front propagation.

In this talk, I will present a proof that the measures of level sets of the Riemann zeta function have Gaussian tail, up to a constant C, for values in suitable regimes, conditionally on the Riemann Hypothesis. As a corollary, we recover the best-known bounds on the moments on the critical line. The proof relies on the recursive scheme of prior work with Bourgade & Radziwill that is inspired by a random walk heuristic. It also combines ideas of Soundararajan and Harper. We will discuss possible improvements to the constant C as well as the connections with the Keating-Snaith Conjecture from Random Matrix Theory for the optimal constant.

This is joint work with Emma Bailey & Asher Roberts and Nathan Creighton.

Given a countable group $G$ equipped with a probability measure $\mu$, one can define a random walk on $G$ as the Markov process whose increments are iid sampled by $\mu$. The large-scale properties of this random walk can exhibit a plethora of exotic behaviours, relating to the algebraic and geometric structure of $G$.

One way of capturing the large-scale behaviour is via the Poisson boundary of $(G,\mu)$, a canonical (abstract) measure space associated with the Markov chain. This object has been studied intensely over the decades. In particular, mathematicians have found concrete 'realizations' of this measure space as topological boundaries for the group (i.e. the Gromov boundary of a hyperbolic group). In 1983, Kaimanovich and Vershik asked whether the Poisson boundary can always be realized in this way.

I will describe the complete resolution of this problem in the negative. This is joint work with Joshua Frisch.

Random band matrices are Hermitian matrices with random entries supported in a band of width $W$ around the diagonal. The eigenfunctions of such matrices are expected to decay exponentially at the scale $W^2$ in dimension one, and $\exp(CW^2)$ in dimension two. Remarkably, the same scaling is expected for the cycle lengths in various models of random permutations where points are typically displaced by distances of order $W$. In this talk I will discuss some recent progress on these problems, some connections, and some open questions.

Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.

In this talk I will explain some ideas behind the proof of rotational invariance of the FK-percolation model and more recently the computation of critical exponents of planar FK model with q=4. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz. We will then explore the connection to the six-vertex model further to derive critical exponents of the FK model with q=4.

Based on joint works with Hong-Bin Chen, Hugo Duminil-Copin, Tiancheng He, François Jacopin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, Tatiana Tikhonovskaia, and Jiaming Xia.

Weak optimal transport is a relaxation of classical optimal transport in which the cost depends nonlinearly on the coupling through its disintegration. This framework is motivated by applications in transport inequalities, entropic optimal transport, and martingale optimal transport. 

In this talk, we propose a gradient flow approach to weak optimal transport in the adapted Wasserstein space. We formally derive the corresponding adapted Wasserstein derivative, project it onto the tangent space of couplings, leading to a McKean-Vlasov SDE that characterizes the flow. We establish well-posedness of this dynamic, and prove its convergence to the optimal coupling for the weak optimal transport problem. This talk is based on the joint work with Nathan Sauldubois. 

When studying a non-cooperative stochastic differential game, it is classical to use the mean-field game framework. However, this approach has its limitations when the particles interact through a random graph. This is because we expect the heterogeneity of the system to be retained in the large population limit. In this talk, we consider a non-cooperative stochastic differential game with player interactions modeled through a sequence of graphs converging to a graphon. We show that under standard regularity conditions and dissipativity of the drift of the state process, when the number of players goes to infinity, the game converges to a general graphon game. The limit is quantitative and based on a new form of propagation of chaos for interacting FBSDEs on graphs. Further, when the cost function satisfies the Lasry-Lions monotonicity condition, we propose an existence and uniqueness result for graphon games. Our proof relies on investigating a system of uncountably many McKean-Vlasov type BSDEs driven by a continuum of Brownian motions. We study this BSDE in a specific probability framework called a Fubini extension and discuss its well-posedness, characterize the graphon mean-field equilibrium and show its existence and uniqueness. The existence proof is based on a new application of the continuation method. Based on joint works with Dylan Possamaï and Ludovic Tangpi.

We identify the scaling limit of full-plane Kadanoff-Ceva fermions on generic, non-degenerate s-embeddings. In this broad setting, the scaling limits are described in terms of solutions to conjugate Beltrami equations with prescribed singularities. For the underlying Ising model, this leads to the scaling limit of the energy-energy correlations connecting the scaling limits of (near-)critical planar Ising models to Green kernels of uniformly elliptic operators and quasiconformal mappings. For grids approximating bounded domains in the complex plane, we establish, in the scaling regime, the conformal covariance of the energy density on critical doubly periodic graph. Moreover, we prove a connection between the planar Ising energy field in smooth limiting structure and (massive)-holomorphic fermions defined on surfaces of the Minkowski space R^{(2,1)}. These results highlight that the scaling limits of generic (near-)critical Ising models naturally live on a substantially richer conformal structure than the classical Euclidean one.

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