NEPS 2025 Talks and Abstracts

The critical 2-d Stochastic Heat Flow arises as a non-trivial solution of the Stochastic Heat Equation (SHE) and its discretisation via directed polymers at the critical dimension 2 and at a phase transition point. It is a log-correlated field, which is neither Gaussian nor a Gaussian Multiplicative Chaos. We will review the phase transition of the 2-d SHE, describe the main points of the construction of the Critical 2-d SHF and outline some of its features (e.g. singularity, regularity, moments, support etc.) Most of the talk will be based on joint works with Francesco Caravenna and Rongfeng Sun but contributions of other researchers will be presented.

The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified stochastic heat equation, specifically at the critical dimension of two and near the critical temperature. I will present an axiomatic formulation of the SHF as well as its construction based on its moments, and discuss how this formulation can be applied to solve a range of problems.

Stochastic analysis was created by Itô in the 40s and this approach dominated the theory of stochastic processes until the end of the 20th century and beyond. In the last 25 years a series of new ideas have enriched this theory and allowed spectacular progress, notably in the study of stochastic partial differential equations. The main techniques in this new wave of research are rough paths, the sewing lemma, regularity structures, paracontrolled distributions. In this talk I want to review some of the main ideas of this line of research, in particular the new notion of pathwise stochastic integration and the crucial construction of a product between certain classes of (random) Schwartz distributions.

Over the past two decades, the rapid growth of data on real-world networks and their influence on our lives has spurred the development of a wide range of probabilistic models. These models are often motivated by domain-specific questions—for example, how information spreads, or how robust a system is under random failures of nodes and edges. In many such settings, problems that appear static at first glance such as the geometry of minimal spanning trees in strong-disorder regimes, or the size and fluctuations of the largest connected component can be reformulated in terms of dynamic processes where the network evolves over time. This perspective creates a natural bridge to coagulation processes, classical objects with origins in colloidal chemistry. Such connections not only help resolve the specific questions of interest, but also reveal universality principles with a range of models exhibiting the same large network behavior. This talk will survey recent developments at this interface, including universality in critical percolation, its surprising connections to scaling limits of minimal spanning trees via Erdős’s leader problem, and new results on fluctuations of microscopic and macroscopic functionals in inhomogeneous random graphs.

We study particle systems interacting via hitting times on sparsely connected graphs, following the framework of Lacker, Ramanan and Wu (AAP, 2023). We provide general robustness conditions that guarantee the well-posedness of physical solutions to the dynamics, and demonstrate their connections to the dynamic percolation theory. We then analyze the limiting behavior of the particle systems, establishing the continuous dependence of the joint law of the physical solution on the underlying graph structure with respect to local convergence and studying the convergence of the global empirical measure, which extends the general results by Lacker et al. to systems with singular interactions. The model proposed provides a general mathematical framework in continuous time for analyzing systemic risks in large sparsely connected financial networks with a focus on local interactions, featuring instantaneous default cascades.

We’ll discuss dynamical extensions of Fisher-Hartwig asymptotics for Hermitian matrix processes evolving under the stationary Ornstein-Uhlenbeck flow. In this setting, both jump- and root-type singularities in space-time are treated simultaneously. From these asymptotics, we derive convergence of fractional powers of characteristic polynomials (and the exponential eigenvalue counting process) to a two-dimensional Gaussian multiplicative chaos measure in the subcritical regime, providing a time-dependent generalization of known static results. The framework also yields optimal rigidity estimates for non-intersecting Brownian motions.

In a mean-field population in the presence of selfing, the trajectories of gene genealogies even far apart on the genome are coupled by exchangeable fragmentation-coalescence processes in the sense of Berestycki '04. This is a new scaling limit arising from a quenched approach to coalescent theory and relies on a novel characterization of convergence in distribution of random measures on Skorokhod space.

The fluctuations of the passage time in first passage percolation have been and are extensively studied. We show that the non-random fluctuations in planar FPP are at least of order \( \log(n)^\alpha \) for any \( \alpha<1/2 \) under some conditions that are known to be met for a large class of absolutely continuous edge weight distributions. This improves the \( \log(\log(n)) \) bound proven by Nakajima and is the first result showing divergence of the fluctuations for arbitrary directions.

Our proof is an application of recent work by Dembin, Elboim and Peled on the BKS midpoint problem and the development of Mermin-Wagner type estimates.

We consider a SPDE, let's say the one-dimensional multiplicative wave equation equation by a space time white noise. We want to study the behaviour of the law of the couple of random variables (spatial average in \( [-R, R] \), an evaluation of a solution) when the domain of integration R grows to infinity. We use an adaptation of Stein's method to prove the convergence toward the product of a a Gaussian and the law of the solution.

The chemical (intrinsic) 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 strength of the result derives from the generality of the method, which uses the robust incipient infinite cluster constructed in our previous work to enforce a novel decoupling argument that separates neighborhoods of distant pivotal vertices. In addition, this decoupling tool proves to be useful in studying the mass structure of percolation clusters, which is necessary in the steps towards a scaling limit result. This is joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe. The preprint can be found at https://arxiv.org/abs/2509.06236.

The Sine-beta process arises as the bulk scaling limit of the beta-ensembles in random matrix theory. I will present a representation of its pair correlation function for all positive beta via a stochastic differential equation. This allows us to show that the pair correlation function depends continuously on beta and to derive estimates for its asymptotic decay. It also recovers the classical explicit formulas in the special cases when beta is 2 and 4.

We study the implicit regularization effect of early stopping in the training of large neural networks. By formulating the training dynamics as a mean field control problem, we characterize the regularization phenomenon through an energy-dissipation perspective. Our analysis is supported by numerical experiments that illustrate and validate the theoretical findings.

We investigate the number of real zeros of random Weyl polynomials with general coefficients. We determine how the expected number of real roots and their variance depend on the moments of the common coefficient. Our method is based on an Edgeworth expansion for random walks arising from Weyl polynomials, which is of independent interest.

For the supercritical Bernoulli bond percolation on \( \mathbb{Z}^d \) (\(d \geq 2 \)), we give a coupling between the random walk on the infinite cluster and its limit Brownian motion, such that the maximum distance between the paths during \([0,T]\) has a mean of order \(T^{\frac{1}{3}+o(1)}\). The construction of the coupling utilizes the optimal transport tool. The analysis mainly relies on local CLT and the concentration of the cluster density. This partially answers an open question posed by Biskup [Probab. Surv., 8:294-373, 2011]. As a direct application, our result recovers the law of the iterated logarithm proved by Duminil-Copin [arXiv:0809.4380], and further identifies the limit constant.

Motivated by the connection to a probabilistic model of phylogenetic trees introduced by Aldous, we study a recursive sequence known as the harmonic descent chain. While it is known that this sequence converges to an explicit limit \( x \), not much is known about the rate of convergence. We prove the asymptotic convergence rate \( x_n - x = n^{-\gamma_* + o(1)}\) for an implicit exponent \(\gamma_*\). Then, implementing a variant of the contraction method with degenerate limit equation, we deduce central limit theorems for various statistics of the critical beta-splitting random tree. This answers a number of questions of Aldous, Janson, and Pittel.

In this talk, we introduce a generalized Bernoulli process (GBP), a stationary binary sequence that can have long-range dependence. The large-scale behavior of GBP resembles that of the fractional Poisson process, when the covariance function of GBP decays with a power-law. More generally, we show the scaling limits of dependent random walks that follow GBPs with various covariance functions. The limiting processes are continuous time stochastic processes with stationary increments whose correlation decays with an exponential rate, a power law, or an exponentially tempered power law. The limit densities solve a tempered time fractional diffusion equation or time fractional diffusion equation. The second family of Mittag-Leffler distribution and the exponential distribution arise as special cases of the limiting distributions.

In this talk, I'll discuss community detection in the regime where the number of communities \( q \) is growing with the size of the graph \( n\). It turns out that when \(q \ll \sqrt{n}\) the sharp transition for weak recovery remains at the well-known Kesten–Stigum bound. Perhaps surprisingly, we find that when \(q \gg \sqrt{n}\), there is an efficient algorithm for recovery even below the KS bound. We identify a new threshold which we conjecture is the threshold for weak recovery in this regime. Joint work with Elchanan Mossel, Youngtak Sohn, Alex Wein.

The infinity Laplace equation arises as a prototype in calculus of variations in \(L^{\infty}\) norm. A breakthrough by Peres, Schramm, Sheffield, and B. Wilson revealed that this equation can be understood as the value of Tug-of-War stochastic games. In this talk, I will present two variants of tug-of-war type games. First, I will discuss the variational structure of the biased infinity Laplacian, formulated as the absolute minimization of a suitable “biased” slope. As an application, we show that biased infinity harmonic functions are everywhere differentiable. Second, I will introduce tug-of-war with killing, which serves as an analogue of the celebrated Feynman–Kac formula for infinity Laplace-type operator. I will demonstrate that the killing operation has a regularizing effect on the game, and then establish the existence of the game's value as well as the existence and uniqueness of the associated PDE.

For the log-Gamma polymer model, the geometric RSK algorithm was used to produce formulas for the Laplace transform of the partition function. In this work, we generalize gRSK to account for an external field, where there is a bias in one direction. We use similar methods to the work of Bisi and Zygouras to give formulas for the new partition function.

We prove the cutoff phenomenon for the Glauber dynamics of the q-state Potts model on \(\mathbb{Z}^d\) at sufficiently high enough temperature. The mixing time exhibits a sharp transition at \(t_{\mathrm{mix}} =O(\log |\Lambda|)\), with an \(O(1)\) cutoff window. Our analysis builds on the information-percolation framework: by tracing update supports backward in time, we show that influence clusters are subcritical, causing correlations from the initial condition to decay exponentially fast. This establishes the first sharp \(O(1)\) cutoff window result for the Potts model, where the lack of monotonicity compared to the Ising case poses significant technical challenges.”

We are interested in the behavior of the microscopic point process for Coulomb gases at intermediate temperature regimes \( \beta\to 0 \), \(\beta N\to\infty\). The behavior at high temperatures \(\beta N\sim 1\) is well understood (cf. Lambert '21) for general Riesz gases and at intermediate temperatures for Gaussian β-ensembles (Benaych-George, Péché '15), but the intermediate temperature behavior for Coulomb and Riesz gases is not well understood. In our work, we establish convergence of the microscopic point process for the Coulomb gas in \(d=2\) to a Poisson point process for a strictly larger intermediate temperature regime than was previously known. Our approach relies on a precise asymptotic description of the correlation functions and overcrowding estimates due to Thoma '25.

We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Our approach extends Talagrand's covering argument, which is based on certain sojourn time estimates.

The totally asymmetric simple exclusion process is a widely studied interacting particle system where particles hop to the right on a line according to independent rate 1 Poisson clocks, with the constraint that two particles cannot occupy the same space. We consider this process on a finite line segment of length N with open boundaries, so that particles can also enter from the left boundary and exit from the right. It was previously shown by Elboim and Schmid that under high and low density conditions, this process rapidly changes from unmixed to mixed in a critical time window around \(CN\), where \( C \) is an explicit constant—the so-called cutoff phenomenon. We refine this statement to show the exact decay of the total variation distance within the critical time window. Along the way, we obtain an enlightening picture of how the process gradually mixes. This is based on ongoing work with Dominik Schmid.

The Kardar–Parisi–Zhang (KPZ) equation is a ubiquitous model of random growth. We consider the equation in a half space with Neumann boundary condition. This model has seen significant recent attention due to the presence of a “depinning” phase transition and a rich phase diagram for its limiting distributions as the boundary strength varies. While scaling limits for the full-space KPZ equation have been studied extensively, much less is known in the half-space setting. In this talk, I will discuss results on the process-level long-time scaling limit of the half-space KPZ equation under 1:2:3 KPZ scaling. The key to the analysis is the construction of a novel half-space Gibbsian line ensemble, a collection of interacting Brownian curves with pairwise attraction at the boundary, into which the half-space KPZ equation embeds as the top curve. This is based on joint work with Sayan Das.

In the area of Markov chain mixing, during the past few years, there has been a significant effort to refine previously known cutoff phenomenon results, by determining the limit profile of the Markov chain around the cutoff time. In this talk, we consider the high-temperature Ising model on the complete graph (also known as the Curie-Weiss model) and establish the limit profile for the respective Glauber dynamics. The proof seeks to use the fact that once the two-coordinate chain corresponding to the model arrives near the center of mass, its trajectory approximately follows the solution of a two-dimensional stochastic differential equation.

Given a graph sequence \(G_n\) converging in cut metric to a graphon \(W\), we sample \(1\ll N\ll n\) vertices and observe the induced subgraph \(\widetilde G_n\). The empirical (co)degree distribution of \(\widetilde G_n\) defines a random measure whose deviation from its population counterpart captures the sampling noise in higher-order connectivity. We establish a functional CLT for its empirical process, and the proof combines multivariate Stein’s method with a sparse counting lemma for subgraphs of bounded degeneracy. This talk is based on joint work with Sumit Mukherjee.

Lattice Yang-Mills theory provides a well defined mathematical description for the standard model of particle physics. The model assigns edges of the lattice a matrix from a compact matrix Lie group such as \(SU(N)\), \(U(N)\), or \(SO(N)\), and there is a parameter in the model, β, known as the inverse coupling strength. In this talk we summarize recent progress in understanding two fundamental properties for lattice Yang-Mills, mass gap and area law, at strong coupling (small β). In 2023 Shen, Zhu, and Zhu used a dynamical approach to establish a mass gap for \( \beta < c_d N \) for \(SU(N)\) and \(SO(N)\) and a dimensional constant \(c_d\). Then in a sequence of two papers both released this year, Sky Cao and Scott Sheffield and I showed that area law holds for \(U(N)\) at a uniform rate for \(\beta < c_d\), and later that area law holds for some rate for \(\beta < c_d N\) for the groups \(SU(N)\), \(SO(2N)\) and \(U(N)\). Finally in ongoing work the mass gap result of Shen, Zhu, and Zhu is extended to \(U(N)\). All of these works improve upon the classical cluster expansion results of Osterwalder and Seiler from 1978. After explaining the results, there will be a brief discussion of what simplifications occur when \(N\) is large, and what techniques are able to capture the large \(N\) simplification.

Recent work has revealed multiple contour integral formulas for the multi-point distribution of the parabolic Airy process, arising from distinct approaches in integrable probability and random matrix theory. Although these formulas look quite different—one written as a Fredholm determinant, the other as a nested contour integral—they must encode the same underlying law. In this talk, I will explain how these representations can be brought together through a sequence of contour deformations, residue evaluations, and a generalized Andreief identity. ​This correspondence also suggests a unifying algebraic framework that we hope to apply to other Airy-type and KPZ-class processes.

We study the limiting one-point fluctuations of the geodesic in the directed landscape, conditioning on its length tending to infinity. Previous works have shown that when the directed landscape value \(L\) between \((0,0)\) and \((0,1)\) is large, the geodesic stays within a narrow \(O(L^{-1/4})\) strip and, after rescaling, behaves like a Brownian bridge. Moreover, the geodesic length fluctuates on the order of \(L^{1/4}\) with a Gaussian limit. In this talk, we focus on finer fluctuations near the endpoints, conditioned on the directed landscape value \(L\) being large. We identify a critical scaling window \(L^{-3/2}\) : \(L^{-1}\) : \(L^{-1/2}\) for the time, geodesic location, and geodesic length, and show convergence to a nontrivial limiting joint distribution. When the time parameter is sent to infinity, this limiting distribution converges to the joint distribution of two independent Gaussian random variables, consistent with previous results. We also find a surprising connection between this new limiting distribution and the upper-tail field of the KPZ fixed point. This talk is based on joint work with Zhipeng Liu and Chenyang Ma.