Probability Archives for Fall 2025 to Spring 2026
Expanding on Average Random Dynamics
When: Wed, September 3, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Jonathan DeWitt (University of Maryland- College Park) - https://math.umd.edu/~dewitt/
We study exponential mixing for volume preserving random dynamical systems on surfaces. Suppose that (f_1,...,f_m) is a tuple of volume preserving diffeomorphisms of a closed surface M. We now consider the uniform Bernoulli random dynamical system that this tuple generates on M . We assume that this tuple satisfies a condition called being "expanding on average," which means that the log of the norm of unit tangent vectors grows in expectation, where the expectation is taken over all the realizations of the random dynamics. From this assumption we show quenched exponential mixing. (This is joint work with Dmitry Dolgopyat)
Large deviation principle for the Airy point process
When: Wed, September 10, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Chenyang Zhong (Columbia University) -
he Airy point process is a universal determinantal point process and a central object in random matrix theory. It arises from eigenvalues near the soft edge of large random matrix ensembles and the largest parts of random partitions picked from the Plancherel measure. In this talk, I will present a large deviation principle for the Airy point process. The large deviations result resolves a conjecture of Ivan Corwin and Promit Ghosal, and is connected to lower tail large deviations of the KPZ equation. Our result also extends to point processes arising from the stochastic Airy operator.
: Favorite sites for simple random walk in two and more dimensions
When: Wed, September 17, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Izumi Okada (University of Tokyo) - https://sites.google.com/view/izumiokadamath/%E3%83%9B%E3%83%BC%E3%83%A0
Abstract:We consider favorite sites, i.e., sites that achieve the maximal local time for a discrete time simple random walk. We show that the limsup of the number of favorite sites is 3 with probability one in d = 2. We also give sharp asymptotics of the number in higher dimensions. This talk is based on a joint work with Chenxu Hao (Peking), Xinyi Li (Peking) and Yushu Zheng (CAS).
Hyperbolic Anderson equations and Brownian intersection local times
When: Wed, September 24, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Xia Chen (University of Tennessee- Knoxville) - https://web.math.utk.edu/~xchen3/
An idea recently merged from the investigation of hyperbolic Anderson equations is
to represent the chaos expansion of the solution in terms of Brownian intersection local
times. In this talk, I will address effeteness, current state, potentials and challenge about
this method.
Part of the talk comes from the work joined with Yaozhong Hu.
Narrow escape problems and diffusion processes in narrow tubes
When: Wed, October 1, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Wen-Tai Hsu (UMD) - In this talk, we discuss a diffusion process in a narrow tubular domain with reflecting boundary conditions, where the geometry serves as a singular perturbation of an underlying graph in $\mathbb{R}^2$ or $\mathbb{R}^3$. We show that, in the limit, the projected process converges weakly to a diffusion process on the graph, with gluing conditions at the vertices that depend on the relative scales of the neighborhoods. Our analysis relies on a detailed understanding of the narrow escape problem in domains with bottlenecks. In particular, we rigorously derive the asymptotic behavior of the expected escape time, establish the asymptotic exponential distribution of escape times, and obtain exit place estimates, results that may be of independent interest.
The directed landscape is a black noise
When: Wed, October 8, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Shalin Parekh (University of Maine) - https://umaine.edu/mathematics/2014/08/29/shalin-parekh/
Abstract: : The directed landscape is the central object in the Kardar-Parisi-Zhang universality class, and is conjecturally the scaling limit for all models of last passage percolation, directed polymers, exclusion processes, and many other types of interface growth models arising in probability and statistical physics. It was constructed independently by Matetski-Quastel-Remenik and Dauvergne-Ortmann-Virag. In this talk, we discuss a recent result where we show that the directed landscape is a black noise. This roughly means that it is noise-sensitive and cannot be expressed as a random dynamical system driven by Gaussian white noise. In particular, it cannot be written as a stochastic PDE.
Moderate deviations for the capacity of the random walk range
When: Wed, October 15, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Jiyun Park (Stanford University) -
It is known that the capacity of the range of a random walk in n dimensions behaves similarly to the volume of the random walk in n-2 dimensions. In this talk, we extend this analogy to the moderate deviations of the capacity in dimension 5. In particular, we demonstrate that the large deviation principle transitions from a Gaussian tail to a non-Gaussian tail depending on the deviation scale. We also improve previously known results for dimension 4. This is based on joint work with Arka Adhikari.
The maximum of Poissonian log-correlated fields
When: Wed, October 22, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Nick Cook (Duke University) - https://sites.math.duke.edu/~nickcook/
Extreme values of logarithmically correlated fields have been extensively studied in connection with Gaussian multiplicative chaos, random matrices, branching random walks, reaction-diffusion PDE and L-functions. The sharpest results are for Gaussian or nearly-Gaussian fields. On the other hand, characteristic polynomials of sparse random matrices give rise to log-correlated fields with Poissonian tails. In prior work with Zeitouni we obtained the leading order of the maximum for the characteristic polynomial of random permutation matrices. I will discuss a refined result on the maximum for a related class of random trigonometric polynomials, which we find is modeled by the rightmost particle of a branching random walk in random time-environment. Based on work with Haotian Gu (UCLA).
Directed distance on bipolar-oriented triangulations: A path toward the directed Liouville quantum gravity metrics
When: Wed, October 29, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Jacopo Borga (MIT) - https://www.jacopoborga.com/
Last and first passage percolation in two dimensions are classical discrete models of random directed planar Euclidean metrics in the KPZ universality class. Their scaling limit is described by the directed landscape of Dauvergne-Ortmann-Virág.
Random planar maps are classical discrete models of random undirected planar fractal metrics in the LQG universality class. Their scaling limit is described by the (undirected) LQG metric of Ding-Dubédat-Dunlap-Falconet and Gwynne-Miller.
We present recent progress on the study of longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations around a typical edge and is expected to be in the LQG universality class with $\gamma=\sqrt{4/3}$. We first explain the analogies between this model and last and first passage percolation. Then, we construct the Busemann function, which measures directed distance to infinity along a natural interface of the UIBOT. We show that, in the case of longest (resp. shortest) directed paths, this Busemann function converges in the scaling limit to a $2/3$-stable Lévy process (resp. a $4/3$-stable Lévy process). These results imply that in a typical subset of the UIBOT with $n$ edges, longest directed path lengths are of order $n^{3/4}$ and shortest directed path lengths are of order $n^{3/8}$.
We conclude the talk by explaining why these results fit into a program to construct the (longest and shortest) directed LQG metrics, two distinct two-parameter families of random fractal directed metrics which generalize the LQG metric and which could conceivably converge to the directed landscape upon taking an appropriate limit.
Based on joint work with E. Gwynne.
Ferromagnetic Potts measures on large locally tree-like graphs.
When: Wed, November 5, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Amir Dembo (Stanford University) - https://adembo.su.domains/
Fixing integers q, d>2, denote by Q(n,T,B) the ferromagnetic q-Potts measures on graphs G(n),
at temperature T>0 and non-negative external field strength B, where as n grows the uniformly
sparse G(n) of n vertices converge locally to the infinite d-regular tree. I will review a joint work
with Anirban Basak and Allan Sly, showing that the convergence of the Potts free energy
density to its Bethe replica symmetric prediction (which was proved for even d, or when B=0),
yields the local weak convergence of Q(n,T,B) to the corresponding free or wired Potts measure
on the infinite tree. One gets the free versus wired limit, according to which has the larger Potts Bethe
functional value, with mixtures of these two appearing as limit points at the critical temperature
T_c(q,B), where these two values of the Bethe functional coincide. For edge-expander G(n),
we also establish a pure-state decomposition by showing that below the critical temperature,
conditionally on having a dominant color k, the measures Q(n,T,0) converge locally to the
q-Potts measure on the infinite tree, with a boundary wired at color k.
Phase transitions of random constraint satisfaction problems
When: Wed, November 12, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Youngtak Sohn (Brown University) - https://youngtaksohn.github.io/
The framework of constraint satisfaction problems (CSPs) captures many fundamental problems in combinatorics and computer science, such as finding a proper coloring of a graph or solving Boolean satisfiability problems. To study the typical cases of CSPs, statistical physicists have proposed a detailed picture of the solution space for random CSPs based on non-rigorous methods from spin glass theory. In this talk, I will first survey the conjectured rich phase diagrams of random CSPs in the one-step replica symmetry breaking universality class. Then, I will describe mathematical progress in understanding the global and local geometry of solutions, particularly in random regular NAE-SAT problem. This is based on joint work with Danny Nam and Allan Sly.
Geodesics and approximate geodesics in critical 2D first-passage percolation
When: Wed, November 19, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Erik Bates (NCSU) - https://www.ewbates.com/
First-passage percolation on the square lattice is a random growth model in which each edge of Z^2 is assigned an i.i.d. nonnegative weight. The passage time between two points is the smallest total weight of a nearest-neighbor path connecting them, and a path achieving this minimum is called a geodesic. Typically, the number of edges in a geodesic is comparable to the Euclidean distance between its endpoints. However, when the edge-weights take the value 0 with probability exactly 1/2, a strikingly different behavior occurs: geodesics travel primarily on critical clusters of zero-weight edges, whose internal graph distance scales superlinearly with Euclidean distance. Determining the precise degree of this superlinear scaling is a challenging and ongoing endeavor. I will discuss recent progress on this front (joint with David Harper, Xiao Shen, and Evan Sorensen), along with complementary results on a dual problem, where we restrict path lengths and analyze passage times (joint with Jack Hanson and Daniel Slonim).
Disconnection and non-intersection probabilities of Brownian motion on an annulus
When: Wed, December 3, 2025 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Gefei Cai (Peking University) - https://gefei-cai.github.io/
We derive an exact formula for the probability that a Brownian path on an annulus does not disconnect the two boundary components of the annulus. The leading asymptotic behavior of this probability is governed by the disconnection exponent obtained by Lawler-Schramm-Werner (2001) using the connection to Schramm-Loewner evolution (SLE). The derivation of our formula is based on this connection and the coupling with Liouville quantum gravity (LQG), from which we can exactly compute the conformal moduli of random annular domains defined by SLE curves. Using a similar approach, we also derive exact formulas for the non-intersection probabilities of independent Brownian paths on an annulus, as well as extend the result to the case of Brownian loop soup. Based on joint work with X. Fu, X. Sun, and Z. Xie, and upcoming work with Z. Xie.
Invariant measure for open KPZ
When: Wed, January 28, 2026 - 2:00pmWhere: TBA
Speaker: Yu Gu (UMD) -
Abstract: Given an evolving string on [0,1] modeling random surface growth, one may ask: if the slopes at the two boundary points are prescribed, what does a typical realization look like? This question reduces to understanding the equilibrium state of the open KPZ equation. In recent work of Corwin–Knizel, Barraquand–Le Doussal, and Bryc–Kuznetsov–Wang–Wesołowski, this equilibrium state is described explicitly and identified as a resampled Brownian motion.
In this talk, I will present recent joint work with Alex Dunlap and Tommaso Rosati in which we provide a stochastic-analytic derivation of this equilibrium state. Our approach relies on tools such as change of measure, time reversal, Itô’s formula, and the theory of regularity structures.
Random covers of hyperbolic surfaces
When: Fri, February 6, 2026 - 1:30pmWhere: Math 3206
Speaker: Ramon van Handel (Princeton University) - https://web.math.princeton.edu/~rvan/
Abstract: It was understood long ago that the first nonzero eigenvalue of the
Laplacian on a closed hyperbolic surface cannot exceed that of the
hyperbolic plane, asymptotically as the diameter goes to infinity. But
until recently, it was not known whether there exist hyperbolic surfaces
that attain this bound. In joint work with Magee and Puder, we show
something much stronger: random covering spaces of an arbitrary
hyperbolic surface have an optimal spectral gap with probability 1-o(1).
I will aim to explain how this question reduces to an unusual problem of
random matrix theory, whose resolution was made possible by recent
advances in the theory of strong convergence.
Identifying the deviator
When: Wed, February 11, 2026 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Eilon Solan (Tel Aviv University) - https://sites.google.com/site/eilonsolanphd/
Abstract: Alice and Bob control a random walk: alternately, each of them flips a fair coin, is supposed to report the outcome, and the random walk advances according to the report. Suppose that the random walk did not return to the origin infinitely often. We suspect that one of Alice and Bob misreported the outcomes of her or his coin. Can we identify the deviator?
More generally, several players are supposed to follow a prescribed profile of strategies (e.g., select each of Right and Left with probability 1/2). If they follow this profile, they will reach a given target (e.g., the random walk returns to the origin infinitely often). We show that if the target is not reached because some player deviates, then an outside observer can identify the deviator. We also construct identification methods in two nontrivial cases.
Joint work with Noga Alon (Princeton and Tel Aviv University), Benjamin Gunby (Rutgers), Xiaoyu He (Georgia Tech), and Eran Shmaya (Stony Brook).
First Passage Percolation in a Product-Type Random Environment
When: Wed, February 18, 2026 - 1:00pmWhere: 1310.0
Speaker: Konstantin Khanin (BIMSA) -
Abstract: We consider a first passage percolation model in dimension 1 + 1 with
potential given by the product of a spatial i.i.d. potential with
symmetric bounded distribution and an independent i.i.d. in time
sequence of signs. We assume that the density of the spatial potential
near the edge of its support behaves as a power, with exponent κ > −1.
We investigate the linear growth rate of the actions of optimal
point-to-point lazy random walk paths as a function of the path slope
and describe the structure of the resulting shape function. It has a
corner at 0 and, although its restriction to positive slopes cannot be
linear, we prove that it has a flat edge near 0 if κ > 0. For optimal
point-to-line paths, we study their actions and locations of favorable
edges that the paths tend to reach and stay at. Under an additional
assumption on the time it takes for the optimal path to reach the
favorable location, we prove that appropriately normalized actions
converge to a limiting distribution that can be viewed as a counterpart
of the Tracy–Widom law. Since the scaling exponent and the limiting
distribution depend only on the parameter κ, our results provide a
description of a new universality class.
The talk is based on a joint work with Yuri Bakhtin, András Mészáros and
Jeremy Voltz.
Mixed boundary identities and sharp deviation estimates in the exponential last-passage percolation
When: Wed, February 18, 2026 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Chris Janjigian (Purdue University) - https://www.math.purdue.edu/~cjanjigi/
Abstract: "On-scale" stretched exponential estimates for fluctuations of models within the Kardar-Parisi-Zhang universality class have played a particularly important role in mathematical work seeking to make physically motivated heuristic arguments about random growth models rigorous.
This talk will explain a simple coupling argument which recovers a moment generating function identity originally discovered by Rains using integrable probability techniques in the context of last-passage percolation. This coupling approach has proven to generalize significantly since it was first introduced about five years ago. The talk will then discuss how identities of this type can be used to derive the type of stretched exponential bounds mentioned above and when these bounds are sharp. The discussion will highlight some recent and in-progress improvements and extensions of older work.
Based on joint work with Elnur Emrah and Timo Seppäläinen.
Anomalous diffusion for Brownian motion with random drift
When: Wed, February 25, 2026 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Ahmed Bou-Rabee (UPenn) - https://nitromannitol.github.io/
Abstract: A Brownian particle subject to a random, divergence-free drift will have enhanced diffusion. The correlation structure of the drift determines the strength of the diffusion and there is a critical threshold, bordering the diffusive and superdiffusive regimes. Physicists have long expected logarithmic-type superdiffusivity at the threshold and algebraic superdiffusion above it. I will discuss recent [arXiv: 2601.22142, arXiv:2404.01115] and ongoing work with Scott Armstrong and Tuomo Kuusi in which we address these problems using techniques from the theory of stochastic homogenization.
Log-concavity and Tracy-Widom universality
When: Wed, March 25, 2026 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Mokshay Madiman (University of Delaware) - https://mokshaymadiman.wordpress.com/
Abstract: The Tracy-Widom (TW) distributions, indexed by a positive parameter b, arise as limiting distributions in many different probabilistic models including random matrices and the longest increasing subsequences of random permutations, thus exemplifying a universality phenomenon in probability. However, they are rather mysterious, with no explicit form for their densities or characteristic functions, and little has been proved analytically about their properties. Although simulations suggest that the TW distributions are log-concave, the only (partial) result in this direction is a theorem of Percy Deift that the TW distribution with parameter 2 is log-concave on the positive real line. We settle this as well as several related questions: Not only are all TW distributions shown to be log-concave, we do this by establishing log-concavity of certain pre-limit distributions, including the largest eigenvalues of Gaussian beta-ensembles and the Poissonized Plancherel measure on Young diagrams that arises in the representation theory of the symmetric group. In particular, one consequence of our results is that a Poissonized version of a 2008 conjecture of W.Y.C.Chen— asserting that for any fixed natural number N, the number of permutations in the symmetric group S(N) that have a longest increasing subsequence of length K, is a log-concave sequence in K — is true. The talk is based on joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.
Transportation cost inequalities for singular SPDEs
When: Wed, April 1, 2026 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Ryoji Takano (Tokyo Metropolitan University) - https://sites.google.com/view/ryojitakano/home/
Abstract: This talk discusses transportation cost inequalities (TCIs) for the laws of solutions to singular stochastic partial differential equations. A TCI bounds a (generalized) Wasserstein distance in terms of relative entropy and yields concentration of measure on metric spaces. The main technical difficulty is to control the norm of the BPHZ model constructing the solution in terms of a suitable noise norm via a local Hölder-type estimate. To obtain such an estimate, we incorporate an additional integrability structure into the regularity-structure framework and apply a spectral gap inequality. This talk is based on joint work with Ismaël Bailleul (Université de Bretagne Occidentale) and Masato Hoshino (Institute of Science Tokyo).
Random attractors and nonergodic attractors for diffusions with degeneracies
When: Wed, April 8, 2026 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Yuri Bakhtin (NYU) -
Abstract: I will talk about diffusions with multiple invariant
manifolds of varying dimensions and thus with varying degrees of
degeneracy. Each of the invariant manifolds may carry an invariant
measure. The long-term statistical properties of such a system may be
governed by one or more such invariant measures and transitions
between them. The resulting nonergodic intermittent averaging cannot
be described by the classical ergodic theory. This is joint work with
Renaud Raquepas and Lai-Sang Young.
Percolation and first-passage percolation on logarithmic subgraphs of Z^2
When: Wed, April 15, 2026 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Michael Damron (Georgia Tech) -
Abstract: In two-dimensional Bernoulli percolation, we declare each edge of the square grid Z^2 to be open with probability p or closed with probability 1-p, independently from edge to edge. There is a critical value p_c = 1/2, such that for p < p_c, all components of open edges are finite, and for p > p_c, there is a unique infinite component of open edges. In ’83, Grimmett introduced the following variant. Let f be a nonnegative real function on [0,\infty), and consider the subgraph G_f of Z^2 induced by the edges between the positive first coordinate axis and the graph of f. Grimmett found that if f(u) \sim a \log u as u \to \infty, the critical value p_c(f) for percolation on G_f equals a specific function of a only. In ’86, Chayes-Chayes considered the function f(u) = a \log(1+u) + b \log(1+\log(1+u)) and showed that if b > 2a, then the percolation G_f has an infinite open component at the critical point (i.e., a discontinuous phase transition). In joint work with Wai-Kit Lam, we prove that the phase transition is discontinuous if and only if b > a, and we compute sharp asymptotics for all p, a, and b of the expected passage time in G_f from the origin to the vertical line x=n in the related first-passage percolation model, improving results of Ahlberg. We also find asymptotics for the variance and a central limit theorem.
The Resolvent Method in Complex Dynamics- with a first application on building a bridge between Schramm-Loewner Evolutions and Random Matrix Theory
When: Wed, April 29, 2026 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Vlad Mărgărint (University of North Carolina, Charlotte) - https://margarintvlad.com/
Abstract: Loewner Dynamics is a tool used in many areas of research such as the study of growth processes in the Complex Plane, Schramm-Loewner Evolutions (SLE) and more. I will describe a newly introduced toolbox that studies the Loewner Dynamics via resolvents of matrices. The method works for both deterministic and random matrices. In the random case, one of the first applications of this method, allows us to connect two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory (RMT). This machinery opens new avenues of research that allow the use of techniques from one field to another. In addition, the regular dynamics in the matrix space provides an alternative to the singular driver dynamics of SLE theory. I will describe the first application of our technique, and a short overview of interactions of our method with other areas of active research, which I aim to explore in a sequence of separate recent projects with collaborators from Europe, Asia and North America. More at https://margarintvlad.com/
Asymptotics of the Resistance of the Critical Series-Parallel Graph via Parabolic PDE Theory
When: Wed, May 6, 2026 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Peter Morfe (PSU) -
Abstract: 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.