Title & Abstract

Jae-Hwan Choi:  Well-posedness and large deviations of underdamped Langevin dynamics with Irregular potentials and its application to the Vlasov-Poisson-Fokker-Planck system

Abstract.  Underdamped Langevin dynamics with Sobolev potentials serve as a critical tool for studying the particle systems satisfying the Vlasov-Poisson-Fokker-Planck system. In this talk, I will present results on the well-posedness and large deviation principles for underdamped Langevin dynamics with Sobolev potentials. This presentation is based on ongoing joint work with Chanwoo Kim, Dohyun Kwon, and Jinsol Seo.

Yoochan Han:  Spectral properties and weak detection in stochastic block models

Abstract.  We consider the spectral properties of balanced stochastic block models of which the average degree grows slower than the number of nodes (sparse regime) or proportional to it (dense regime). For both regimes, we prove a phase transition of the extreme eigenvalues of SBM at the Kesten-Stigum threshold. We also prove the central limit theorem for the linear spectral statistics for both regimes. We propose a hypothesis test for determining the presence of communities of the graph, based on the central limit theorem for the linear spectral statistics.

Kohei Hayashi:  Characterization of gradient condition for asymmetric partial exclusion processes and their scaling limits

Abstract.  We consider partial exclusion processes (PEPs) on the one-dimensional square lattice, that is, a system of interacting particles where each particle random walks according to a jump rate satisfying an exclusion rule that allows up to a certain number of particles can exist on each site. Particularly, we assume that the jump rate is given as a product of two functions depending on occupation variables on the original and target sites. Our interest is to study the limiting behavior, especially to derive some macroscopic PDEs by means of (fluctuating) hydrodynamics, of fluctuation fields associated with PEPs, starting from an invariant measure. The so-called gradient condition, meaning that the symmetric part of the instantaneous current is written in a gradient form, and that the invariant measures are given as a product measure is technically crucial. Our first main result is to clarify the relationship between these two conditions, and we show that the gradient condition and the existence of product invariant measures are mutually equivalent, provided the jump rate is given in the above simple form, as it is imposed in most of the literature, and the dynamics is asymmetric. Moreover, when the width of the lattice tends to zero and the process is accelerated in diffusive time-scaling, we show that the family of fluctuation fields converges to the stationary energy solution of the stochastic Burgers equation (SBE), under the setting that the jump rate to the right neighboring site is a bit larger than the one to the left side, of which discrepancy is given as square root of the width of the underlying lattice. This fills the gap at the level of universality of SBE since it has been proved for the exclusion process (a special case of PEP) and for the zero-range process.

Minjoon Kim:  Classification of 1-D stochastic dynamics with jump rates given by rational functions

Abstract.  Continuous-time Markov chains on non-negative integers with multiple jump sizes can be used for population models with bursty production and decay. In biology, it is common to assume the jump rates to be polynomials, or more generally, rational functions. Under these assumptions, we fully classify the dynamical features such as explosivity, recurrence, positive recurrence, and exponential ergodicity by the maximal degree of the jump rates, mean drift, and variance. We also show some applications to mass-action kinetics and Michaelis-Menten kinetics, which are frequently used in biology.

Taegyun Kim:  Fluctuations of the free energy of the spherical Sherrington–Kirkpatrick model with heavy-tailed interaction

Abstract.  We consider the 2-spin spherical Sherrington–Kirkpatrick model without external magnetic field where the interactions between the spins are given as random variables with heavy-tailed distribution. We show that the free energy exhibits a sharp phase transition depending on the location of the largest eigenvalue of the interaction matrix. We also prove the order of the limiting free energy and the limiting distribution of the fluctuation of the free energy for both regimes. This is joint work with Ji oon Lee.

Jaehun Lee:  Spectral heat contents for Levy processes

Abstract.  For a process X and domain D, the spectral heat content represents the total heat remains in D at time t with Dirichlet boundary condition. It is known that when X is an isotropic stable process with index between 0 and 2, the asymtotic behavior of spectral heat content is given by time sacle with index min(1,1/a). In this talk, we obtain a sharp sufficient condition for Levy processes with index larger than 1 in the domain with inner and outer ball conditions. In particular, the result covers both non-isotropic Levy processes and 1-stable processes. This talk is based on the joint work with Hyunchul Park in SUNY.

Jungkyoung Lee:  Metastability of Langevin dynamics and its application

Abstract.  Metastability is a phenomenon observed in various physical models. While it is interesting in its own right, it also has important applications in problems such as partial differential equations and mixing. In this talk, we will introduce the metastability of Langevin dynamics and present results related to the corresponding parabolic equation, as well as the slow mixing that arises as a consequence of this metastability.

Yong-Woo Lee:  Large deviations of real eigenvalues in the elliptic Ginibre orthogonal ensemble

Abstract.  One of the intriguing properties of the Ginibre orthogonal ensemble (GinOE) is that there is the non-trivial probability of observing real eigenvalues within its spectrum. In this talk, we investigate the large deviation probabilities associated with the number of the real eigenvalues. We study the probabilities of observing an exceptionally large number of real eigenvalues in the GinOE, presenting precise asymptotics and connections to potential theory from a Coulomb gas perspective. Additionally, we generalize our analysis to the elliptic Ginibre orthogonal ensemble which is a one-parameter generalization of the GinOE by modifying its Hermiticity. This talk is based on a joint work with Gernot Akemann and Sung-Soo Byun.

DongJun Min:  Exit Time of Dynamical Systems with Heavy-Tailed Noise

Abstract.  Stochastic dynamical systems driven by heavy-tailed noise have garnered significant attention due to their foundational role in the mathematical frameworks supporting machine learning and artificial intelligence. Previous research has provided results on escape times in multidimensional systems under alpha-stable noise and has examined logarithmic scaling behavior when the noise exhibits exponentially light tailed noise. Furthermore, recent studies have investigated stochastic gradient descent (SGD) within this framework, revealing intriguing connections. In this paper, we aim to bridge existing gaps by presenting a comprehensive analysis of the dynamical spectrum influenced by heavy-tailed noise. Specifically, we explore the phase transition behavior of solutions to stochastic dynamical systems as dictated by varying characteristics of the heavy-tailed noise.

Philippe Moreillon:  Disk counting statistics for the complex and symplectic spherical ensembles

Abstract.  In this talk, I will present new results on the disk counting statistics for the complex and symplectic spherical ensembles. These point processes naturally emerge in random matrix theory and we prove that as the number of points becomes large, the moment generating function associated to any finite family of centered disks is of the form exp(C1 n + C2 \sqrt{n} + C3 + C4/\sqrt{n} +O(1/n)), where the constants C1, C2, C3 and C4 are given in terms of the error function. As we will see, our proof relies on the uniform asymptotics of the incomplete Beta function. This talk is based on an upcoming paper.

Shuta Nakajima:  Maximum of the 2D directed polymer in the subcritical regime and Inhomogeneous Branching Brownian Motion

Abstract.  In this talk, we examine the partition function of the two-dimensional directed polymer, specifically focusing on its maximum over a suitably chosen box in the subcritical regime. We present our main result, which establishes the asymptotic behavior of the maximum of the partition function. We show that this maximum converges in probability to a limit defined by an integral involving branching Brownian motion with inhomogeneous variance. This work is a collaboration with Clement Cosco and Ofer Zeitouni.

Jiwoon Park:  Finite-size scaling via random geometric representation

Abstract.  Finite-size scaling is a classical topic in statistical physics, offering a robust mathematical framework for understanding simulation results with limited resources. It also plays a key role in calculating asymptotic behaviours, such as those observed in scattering particle systems. From the perspective of mathematical physics, finite-size scaling reveals critical phenomena invisible in infinite systems, such as ensemble scaling limits and plateaus.

However, current finite-size scaling theories often lack mathematical rigor, a unifying framework, or have limited applicability. In this talk, I will present geometric proofs of finite-size scaling in a couple of lattice systems. Since geometric methods have long been central to statistical physics on lattice, I expect that this approach will prove to be widely applicable.

Jiyun Park:  Moderate deviations for the capacity of the random walk range

Abstract.  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.

Jeeho Ryu:  Exit time analysis of Kesten's stochastic recurrence equation and SGD

Abstract.  In this lecture, we review classic theory on the Kesten's stochastic recurrence equation which naturally appears in the analysis of various time series such as SGD or GARCH. We then explain our recent result on the escape time analysis on two major regimes which connects the Lyapunov exponent of the process and the escape time from a domain. Then, we try to connect our result to the empirically observed behavior of the time series quoted above.

Junhee Ryu:  Sobolev estimates for SPDEs with nonlocal operators in domains

Abstract.  In this talk, we study a class of stochastic PDEs with nonlocal operators and spatially homogeneous colored noise in domains. The operators we consider are infinitesimal generators of (time-inhomogeneous) Lévy processes, whose Lévy measures are allowed to be very singular. We obtain the well-posedness and regularity of solutions in Sobolev spaces with the admissible spatial dimensions determined by the noise's spatial covariance.

Jinsol Seo:  A general solvability result of the stochastic heat equation in non-smooth domains and its applications

Abstract.  The stochastic heat equation (SHE) is a fundamental SPDE with regularity properties that differ from the classical heat equation, particularly in domain problems. In this talk, we review historical developments related to SHE and its domain problem. I will present a general theorem on the solvability of SHE in non-smooth domains, derived using superharmonic functions, and discuss its application to non-linear stochastic heat equations driven by spatially homogeneous colored noise. This talk is based on a joint work with Jae-Hwan Choi.