Title & Abstract

Gerardo Barrera:  Cutoff phenomenon for the ergodic CIR process

Abstract.  In this talk we investigate the convergence to equilibrium as the noise intensity $\epsilon$ tends to zero for ergodic random systems out of equilibrium driven by multiplicative non-linear noise of the following type \[ d X^{\epsilon}_t(x)=(b-aX^{\epsilon}_t(x))dt+\epsilon \sqrt{X^{\epsilon}_t(x)}dB_t,\quad X_0=x, \] where $x > 0$, $a > 0$ and $b > 0$ are constants, and $(B_t)_{t>0}$ is a one-dimensional standard Brownian motion. More precisely, we establish the occurrence of a profile cutoff phenomenon in the total variation distance and in the renormalized Wasserstein distance when the intensity of the noise tends to zero. Our results include explicit cut-off time, explicit time window, and explicit profile function. Furthermore, asymptotics of the so-called mixing times are given explicitly. This is a joint work with Liliana Esquivel, University of Valle, Colombia.

Johel Beltrán:  Coalescing random walks and the Kingman coalescent model

Abstract.  Given a finite transitive graph and n particles labeled with numbers {1,2,3,...,n}, we place these particles on the set of vertices at random. Then, we let the particles evolve as a system of coalescing random walks: each particle performs a continuous-time simple random walk (SRW) and whenever two particles meet, they merge into one particle which continues to perform a SRW. At each time t, consider the partition P_t of {1,2,3,...,n} induced by the equivalence relation: i~j when particles i and j occupy the same vertex at time t. We show that under a certain condition on the spectral gap of the graph, the Kingman n-coalescent model emerges as a scaling limit for (P_t), as n is fixed and the size of the graph goes to infinity.

Gianmarco Bet:  Detecting a late changepoint in the preferential attachment model

Abstract.  Motivated by the problem of detecting a change in the evolution of a network, we consider the preferential attachment random graph model with a time-dependent attachment function. Our goal is to detect whether the attachment mechanism changed over time, based on a single snapshot of the network at time n and without directly observable information about the dynamics. We cast this question as a hypothesis testing problem, where the null hypothesis is a preferential attachment model with a constant affine attachment parameter \delta_0, and the alternative hypothesis is a preferential attachment model where the affine attachment parameter changes from \delta_0 to \delta_1 at an unknown changepoint time \tau_n. For our analysis we assume that the changepoint occurs close to the observation time of the network, which corresponds to the relevant scenario where we aim to detect the changepoint shortly after it has happened. We present two tests based on the number of vertices with minimal degree. The first test assumes knowledge of \delta_0, while the second does not. We show that these are asymptotically powerful when n−\tau_n \gg n^{1/2}. Furthermore, we prove that the test statistics for both tests are asymptotically normal, allowing for accurate calibration of the tests. If time allows, we will present numerical experiments to illustrate the finite sample test properties.

Tadahisa Funaki:  Interface motion from Glauber-Kawasaki dynamics of non-gradient type

Abstract.  We discuss the derivation of an anisotropic curvature flow from the Glauber-Kawasaki dynamics of non-gradient type. The talk is based on the paper arXiv:2404.18364.  We use the result of Funaki, Gu and Wang, arXiv:2404.12234 for the convergence rate of the diffusion matrix, and that of Funaki and Park, arXiv:2403.01732 for the nonlinear Allen-Cahn equation.

Milton Jara:  Sharp convergence to stationarity for the stochastic Curie-Weiss model

Abstract.  In 2010, Levin, Luczak and Peres showed that the stochastic Curie Weiss model at supercritical inverse temperature $\beta <1$ has mixing time $\frac{1}{2(1-\beta)} n \log n$ and presents cut-off at window size $n$. In order to understand the dependence of the mixing time in this model with respect to the initial configuration  of spins, we introduce the concept of sharp convergence. We show sharp convergence for the stochastic Curie-Weiss model with non-zero magnetic field and also for zero magnetic field and $\beta <1$. As a consequence, we improve the results of Levin, Luczak and Peres to profile cut-off and we show that taking initial configurations with the right magnetization, the mixing time improves to $\frac{1}{2} n \log n$. The proof uses the entropy method and a parabolic version of Stein's method. Joint work with Freddy Hernández (Universidad Nacional de Colombia).

Seonwoo Kim:  Spectral gap of the symmetric inclusion process: Aldous' conjecture and metastability

Abstract.  We consider the symmetric inclusion process on a general finite graph. In the log-concave regime, we establish universal upper and lower bounds for the spectral gap of this process in terms of the spectral gap of the single-particle random walk, thereby verifying the celebrated Aldous' conjecture, originally formulated for the interchange process. Next, in the general non-log-concave regime, we prove that the conjecture does not hold by investigating the so-called metastable regime when the diffusivity constant vanishes in the limit. This talk is based on joint works with Federico Sau.

Hubert Lacoin:  Cutoff phenomenon in nonlinear recombinations

Abstract.  We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial marginals. Our main result reveals a cutoff phenomenon for the total variation distance in both discrete and continuous time. Additionally, we derive the explicit cutoff profiles in the case of monochromatic initial distributions. These profiles are different in the discrete and continuous time settings. The proof leverages a pathwise representation of the solution in terms of a fragmentation process associated to a binary tree. In continuous time, the underlying binary tree is given by a branching random process, thus requiring a more elaborate probabilistic analysis. Joint with P. Caputo (Rome) and C. Labbé (Paris)

Wai-Kit Lam:  On the empirical distribution along a geodesic in first-passage percolation

Abstract.  In first-passage percolation, one assigns i.i.d. nonnegative weights $(t_e)$ to the nearest-neighbor edges of $\mathbb{Z}^d$ and study the induced pseudometric $T$. In this talk, we focus on geodesics (time-minimizing paths) for $T$, and consider the empirical distribution of weights along them. We will first briefly discuss the work by E. Bates, who studied the limit of such measures. We will then talk about a joint work with M. Damron, J. Hanson, C. Janjigian and X. Shen, in which we give estimates on the empirical distribution. One of our main results shows that the tail of the expected empirical distribution along a geodesic is lighter than the original weight distribution by an exponential factor. We also give a lower bound for the tail, and provide estimates of the empirical distribution for more general sets.

Claudio Landim:  Fluctuation in Glauber-Kawasaki dynamics

Abstract.  We find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ in $d\leq 2$. In the one dimensional case, the field limit is given by $e (v_1) B_t$ where $B_t$ is a Brownian motion and $e$ is the normalized derivative of a decreasing `standing wave' solution $\phi$ of $\partial^2_{v_1} \phi - V'(\phi)=0$ on $R$, where $V'$ is the homogenization of the Glauber rates. In two dimensions, the limit is $e(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation.

Jung-Kyoung Lee:  Metastability of Langevin dynamics and parabolic equations

Abstract.  Langevin dynamics exhibits metastability when a potential with multiple equilibria is present. The metastability of the dynamics is closely related to the behavior of a solution to the parabolic equation, whose differential operator is given by the infinitesimal generator of Langevin dynamics. Thanks to the recently developed resolvent approach, we can fully understand the metastability of Langevin dynamics under the assumption that the process has a Gibbs invariant distribution. In this talk, we introduce the resolvent approach as a key method for understanding metastability, and present the results on the metastability of Langevin dynamics. This is joint with Claudio Landim(IMPA) and Insuk Seo(Seoul National University).

Xinyi Li:  Favorite Sites for Simple Random Walk in Two and More Dimensions

Abstract.  In this talk, we consider the evolution of favorite sites on the trace of a discrete-time simple random walk on Z^d. For d=2, we show that limsup of the number of favorite sites is almost surely three. For d≥3, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erdős and Révész (1987). Joint work with Chenxu HAO (Peking University), Izumi OKADA (Chiba University) and Yushu ZHENG (CAS-AMSS).

Mauro Mariani:  Homology of diffusion processes: asymptotic bounds and rigidity

Abstract.  I will discuss rigidity results related to the fluctuations of random homologies associated with diffusion processes on compact Riemannian manifolds. I will focus on the following problem: what can we deduce about the diffusion process if we only observe the law of the number of windings it performs around the holes of a manifold? For instance the homology of a Brownian Motion on a flat torus is Gaussian and its law can be computed explicitly. Which diffusion processes (and on which manifolds) have the same asymptotic behavior of their random homology? As a byproduct, I will show a counterexample to the folklore knowledge "non-reversible converges faster (than reversible)". Joint with A.Galkin.

Kyeongsik Nam:  Critical last passage percolation

Abstract.  Last passage percolation (LPP) is a model of a random metric space where the main observable is a directed path evolving in a random environment. When the environment has light tails, the fluctuations of LPP are predicted to be explained by the Kardar-Parisi-Zhang (KPZ) universality theory. However, the KPZ theory is not expected to apply for many natural environments, particularly "critical" ones exhibiting a hierarchical structure. In this talk, I will talk about such LPP with an inverse quadratic tail decay distribution which is conjectured to be the critical point for the validity of the KPZ scaling relation. As a byproduct, I will mention the resolution of a long-standing question of Martin concerning necessary and sufficient conditions for the linear growth of the LPP energy.

Maximilian Nitzschner:  Directed polymers on supercritical percolation clusters

Abstract.  In this talk, we consider a model of directed polymers defined on a typical realization of an infinite clusters of supercritical Bernoulli bond percolation in dimensions d ≥ 3. We prove that for such realizations and for every strictly positive value of the inverse temperature, the polymer is in a strong disorder regime, thus answering a question from Cosco, Seroussi, and Zeitouni. The proof is based on a utilization of a fractional moment calculation, as well as a large deviation-type estimate on the number of effectively one-dimensional tubes present in the intersection of a large box and the supercritical cluster.

Jiwoon Park:  Random geometric representation and finite-size scaling

Abstract.  Finite-size scaling theory investigates the critical phenomena of statistical physics models in finite volumes, providing insights relevant to both laboratory systems and simulations. It serves as a bridge between theoretical predictions and real-world systems. Traditionally, scaling theories like Landau theory and renormalisation theory have been applied to finite-size scaling due to their successes in statistical physics and quantum field theory. However, in the context of finite-size scaling, these methods often rely on artificial assumptions, such as the introduction of the `qoppa' exponent or a correlation length exceeding the system's size. In contrast, recent advances in stochastic geometric representations within the mathematical physics community offer a promising alternative. By drawing analogies with the random walk representation of the Gaussian free field under periodic boundary conditions, a natural framework for finite-size scaling emerges for models above the upper critical dimension. This framework is grounded in a simple inequality observed widely across statistical physics, providing a more robust and intuitive foundation for the theory.

Alejandro Ramirez:  Weak-strong disorder and KPZ universality in undirected models

Abstract.  The random walk in random environment model and the planar stochastic heat equations, are examples of models of undirected movement (where sites can be visited multiple times) in random environment. The standard weak-strong disorder phase transition of the directed counterparts, and features of the KPZ universality class, are expected to appear. We present a recent result showing the existence of a weak-strong disorder phase transition for (undirected) random walk in random environment in dimensions d>=4, in particular the behavior of the point to point partition function from 0 to 0, and the appearance of the KPZ fluctuations for the planar stochastic heat equation.

Makiko Sasada:  Scaling limits for interacting particle systems on crystal lattices

Abstract.  A crystal lattice is a graph with periodic structures such as Euclidean, triangular, hexagonal, and diamond lattices. On this, we consider general interacting particle systems such as exclusion processes, generalized exclusion processes, multi-species exclusion processes and study their scaling limits. As a special case, simple exclusion processes with jump rates that vary with period 2 on Z have long been investigated, and it is known that the hydrodynamic limit is a linear diffusion equation, although of the non-gradient type. Also, there are already known results for the hydrodynamic limits in the case of simple exclusion processes and zero range processes by using harmonic embedding of lattices. On the other hand, for more general models such as generalized exclusion processes, first of all, even the notion of "gradient model" cannot be naively formulated. Therefore, results for all but the above special models were not known. Recently we have formulated the problem under some general assumptions and identified how the gradient-type and non-gradient-type models, as well as the gradient replacement technique should be understood in this setting. As a main result, we proved the decomposition theorem of germs of closed forms, which is a key for the scaling limits for non-gradient models. This is a joint work with Kenichi Bannai.

Sunder Sethuraman:  An spde as a hydrodynamic limit in a random environment

Abstract.  We consider a one dimensional zero-range interacting particle system subject to a`Sinai'-type random environment.  In this model, we derive in a `two-step' approach, involving regularization of the random environment, that the hydrodynamic limit is a certain nonlinear singular spde.  Connections with Brox and related diffusions for the scaled limit of a tagged particle in the system, will also be discussed. Based on joint works: C. Landim et al (2023), T. Funaki et al (2021), and one in progress with C. Landim and M. Huidani.

Hayate Suda:  Scaling limits of a tagged soliton in the randomized box-ball system

Abstract.  The box-ball system (BBS) is a cellular automaton that exhibits the solitonic behavior. In recent years, with the rapid progress in the study of the hydrodynamics of integrable systems, there has been a growing interest in BBS with random initial distribution. In this talk, we consider the scaling limits for a tagged soliton in the BBS starting from certain stationary distribution.  This talk is based on a joint work with Stefano Olla and Makiko Sasada.

Kenkichi Tsunoda:  Incompressible limit for a weakly asymmetric simple exclusion process with collision

Abstract.  We consider in this talk the so-called incompressible limit for a weakly asymmetric simple exclusion process (WASEP) with collision. One of the fundamental questions in mathematical physics is the derivation of the master equation of fluid such as the Burgers equation or the Navier–Stokes equation. This model has been introduced by Esposito, Marra and Yau (1996) and they have derived the Navier–Stokes equation as an incompressible limit in dimensions strictly larger than two. The derivation of these equations in low dimension is achieved only from a lattice gas which admits mesoscopically long jumps. In the previous work, Jara, Landim and Tsunoda (2021), we considered the WASEP without collision. We discuss in this talk the incompressible limit for the WASEP with collision. This talk is based on joint work with Patrick van Meurs (Kanazawa University) and Lu Xu (Gran Sasso Science Institute). available at arXiv:2402.10375

Linjie Zhao:  Moderate deviation principles for a reaction-diffusion model in non-equilibrium

Abstract.  We consider moderate deviations from the hydrodynamic limits of a reaction-diffusion model. The process is defined as the superposition of the symmetric simple exclusion process with Glauber dynamics. When the process starts from a product measure with a constant density, a non-equilibrium measure for the process, we prove that the rescaled density fluctuation field satisfies the moderate deviation principle. When time permits, we will also outline the main ideas of the proof.