Inhyeok Choi (KIAS)
Title: Regularity of the escape rate and the asymptotic entropy for negatively curved groups
Abstract: Given a discrete group $G$ with a probability measure $\mu$, one can study the asymptotics of the n-fold convolution of $\mu$. This naturally generalizes classical random walks on the integers. There are two quantities that captures the evolution of the $\mu$-random walk, namely, the escape rate and the asymptotic entropy. By the work of Ledrappier, Mathieu and Gouëzel, these quantities are known to depend analytically on $\mu$ when $G$ is word-hyperbolic. In this talk, I will discuss regularity of these quantities for other groups including mapping class groups and cubical groups. Part of these results was independently observed by Anna Erschler and Joshua Frisch.
Jae-Hwan Choi (KIAS)
Title: Boundedness of Nonlocal Operators with Variable Coefficients in Weighted Lebesgue Spaces
Abstract: In this presentation, we study the boundedness of infinitesimal generators associated with subordinated Brownian motions with variable coefficients in weighted Lebesgue spaces. We assume weak scaling conditions on the principal part of the jump kernels and impose only minimal regularity assumptions on the coefficients. Our approach relies on tools from probabilistic potential theory, which allow us to obtain the key estimates required for the analysis.
Masato Hoshino (Institute of Science Tokyo)
Title: Transportation cost inequalities for singular SPDEs
Abstract: We prove that the laws of the BPHZ random models satisfy some transportation cost inequalities in the full subcritical regime if there is no variance blowup and the law of the noise is translation invariant and satisfies some transportation cost inequality. As a consequence, the laws of a number of solutions to some singular stochastic partial differential equations also satisfy some transportation cost inequalities. This is in particular the case of the $\Phi^4_{4-\epsilon}$ measures over the 4-dimensional torus, for all $\varepsilon>0$.
This talk is based on a joint work with Isma\"el Bailleul (Universit\'e de Bretagne Occidentale) and Ryoji Takano (The University of Osaka).
Title: From directed polymers to spatial corrected KPZ equation in any dimensions
Abstract: I will present a joint work with Dr. Meng Wang on the convergence from directed polymers to spatial corrected KPZ equation. It is well-studied about the convergence of directed polymers to stochastic heat equation in the process level in one spatial dimension. Since the Cole-Hopf transformation can relate stochastic heat equation to KPZ equation, we expect that the same transformation of the directed polymer will converge to the KPZ equation. One of the main difficulties is the existence of negative moments, which is always a difficult problem in probability theory. In this talk I will present our recent progress on this negative moment problem. In fact, much of our effort is devoted to the extension to multi-dimension as well.
Hong Chang Ji (University of Wisconsin–Madison)
Title: Spectral estimators in general linear models with invariant designs
Abstract: General linear model (GLM) concerns the statistical problem of estimating a vector x from the vector of measurements $y=Ax+e$, where $A$ is a given design matrix whose rows correspond to individual measurements and $e$ represents errors in measurements. In this talk, we consider GLM with the law of $A$ invariant under right-rotation. We present a spectral estimator for this model, that is, the principal eigenvector of a matrix constructed from A and y achieving positive overlap with $x$, i.e., 'weakly' recovering $x$. We also present the performance guarantee of our estimator in terms of a lower bound for the sample complexity, that is, the number of samples divided by the number of parameters. Our lower bound for the sample complexity matches with the conjectured (obtained by the replica method) threshold in Zdeborová et al, 2020 below which any polynomial-time algorithm cannot recover $x$. Along the proof, we show a new result on strong asymptotic freeness of orthogonal matrices, that is of independent, mathematical interest. If time permits, we will also discuss a new random matrix ensemble, pseudo-Hermitian matrices, that arises in the same context.
Kunwoo Kim (POSTECH)
Title: Invariance principle for stochastic reaction-diffusion equations
Abstract: We establish a notion of universality for the parabolic Anderson model via an invariance principle for a wide family of parabolic stochastic partial differential equations. We then use this invariance principle in order to provide an asymptotic theory for a wide class of non-linear SPDEs. This is based on joint work with Davar Khoshnevisan and Carl Mueller.
Aro Lee (KAIST)
Title: Fluctuations of the Largest Eigenvalues of Transformed Spiked Wigner Matrices
Abstract: We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik–Ben Arous–P´ech´e (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy– Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes. This is joint work with Ji Oon Lee.
Jungkyoung Lee (KIAS)
Title: Mixing of the Curie-Weiss-Potts Model
Abstract: Mixing refers to the convergence of the distribution of a stochastic dynamical system to its stationary distribution. In the high-temperature regime, the Glauber dynamics of the Curie–Weiss–Potts model exhibits fast mixing, which arises from the concentration of the stationary distribution around a unique equilibrium. In contrast, in the low-temperature regime, the presence of multiple equilibria leads to slow mixing, since transitions between distinct equilibria are required for the system to reach stationarity. In this talk, we discuss the slow mixing of the Glauber dynamics for the Curie–Weiss–Potts model in the low-temperature regime.
This talk is based on joint with Seonwoo Kim (Yonsei University).
Yong-Woo Lee (Seoul National University)
Title: Moderate-to-large deviations for real eigenvalues for the real Ginibre ensembles
Abstract: In this talk, we discuss number statistics of real eigenvalues for the real elliptic Ginibre ensembles beyond the central limit theorem regime. We present explicit rate functions for moderate and large deviation behaviors which characterize non-Gaussian fluctuations of the number statistics. In particular, the rate functions recover the known asymptotes in extreme cases where the number of real eigenvalues is either exceptionally small or exceptionally large. In the weak asymmetry, the rate functions provide a full characterization of the distribution of the number of real eigenvalues. This talk is based on joint work with Sung-Soo Byun, Jonas Jalowy and Gregory Schehr.
Yuchen Liao (University of Science and Technology of China)
Title: Space-time joint laws of the KPZ fixed point
Abstract: The KPZ fixed point, first constructed by Matetski-Quastel-Remenik, is a universal scaling limit for the so-called Kardar-Parisi-Zhang universality classa large family of random growth models in 1+1 (space + time) dimension. It can be described as a Markov process on the space of height functions with explicit transition probabilities (fixed time spatial laws).
In this talk I will introduce this object and discuss the integrability around it. In particular I will show some new formulas describing the space-time joint laws of this two-dimensional random fields and discuss some applications of the complicated formulas. Time permitting, I will also discuss on-going work on their periodic analogs.
Insuk Seo & Jeeho Ryu (Seoul National University)
Title: Convergence rate of CFR+ Algorithm
Abstract: Counterfactual Regret Minimization (CFR) is a non-Markov Chain Monte Carlo type algorithm that is known to converge to a stochastic Nash equilibrium strategy in two-player games. The algorithm is designed so that two artificial players repeatedly engage in self-play such a way that the strategy of each player converges to the stochastic Nash equilibrium of the given game. It is well known that the convergence rate of CFR is $O(n^{-1/2})$.
CFR has served as the fundamental algorithm for enabling AI systems to simulate optimal poker strategies, commonly referred to as Game Theoretic Optimal (GTO) strategies. In fact, GTO strategies have not been generated using the original CFR algorithm, but rather a modified variant known as CFR+. This modification was motivated by the expectation that CFR+ achieves a faster convergence rate of $O(n^{-3/4})$ compared to the rate $O(n^{-1/2})$ of CFR. Despite this belief, a rigorous proof of such an improved convergence rate was an open problem.
In this talk, we discuss our result proving that the convergence speed of CFR+ is not $O(n^{-3/4})$.
Kihoon Seong (SLMath)
Title: Sine–Gordon measures, homotopy classes, and topological soliton manifold
Abstract: I will discuss the sine–Gordon QFT, which lies in the same universality class as the Coulomb gas and the XY model, and exhibits the BKT phase transition (Nobel prize 2016). The sine–Gordon action admits infinitely many homotopy classes. For each class, we construct the corresponding Gibbs measure and analyze its behavior in the joint low-temperature and infinite-volume limit.
Surprisingly, even though the action has no energy minimizers in homotopy classes with high topological charge, the Gibbs measure nevertheless concentrates and exhibits Ornstein–Uhlenbeck fluctuations around the multi-soliton manifold, also known as the moduli space. I will then discuss the geometry of this multi-soliton manifold, including how solitons are arranged.
Based on joint work with Hao Shen and Philippe Sosoe.
Yuanyuan Xu (Chinese Academy of Sciences)
Title: Extremal eigenvalues of large non-Hermitian random matrices
Abstract: We will report recent progress on the extremal eigenvalues of a large random matrix with i.i.d. entries. Beyond the radius of the celebrated circular law, we will show that the properly normalized largest eigenvalue converges to a Gumbel distribution, irrespective of the distribution of the matrix entries. Moreover, we also prove that the argument of the largest eigenvalue is uniform on the unit circle and that the extremal eigenvalues form an inhomogeneous Poisson point process on the complex plane. Finally, we will also present several deviation estimates on the extremal eigenvalues of non-Hermitian random matrices.
Seoyeon Yang (Princeton University)
Title: Cutoff with an $O(1)$ window for Potts Glauber Dynamics on $\mathbb{Z}^d$ at High Temperature
Abstract: We prove sharp cutoff with an $O(1)$ window for the continuous-time single-site Glauber dynamics of the ferromagnetic $q$-state Potts model on the discrete torus $\Lambda=(\mathbb{Z}/n\mathbb{Z})^d$ at sufficiently high temperature. There exists $\beta_0(d,q)>0$ such that for all $\beta<\beta_0(d,q)$, the dynamics exhibits a \emph{cutoff} phenomenon, a sharp transition in total-variation distance: there is an explicit constant $\kappa=\kappa(d,q,\beta)\in(-1,0)$ for which
\[ t_{\mathrm{mix}} (\varepsilon)=\frac{1}{2(1+\kappa)}\ \log|\Lambda|+O(1), \]
for every fixed $\varepsilon\in(0,1)$, and the $O(1)$ window is independent of $n$. This appears to be the first $O(1)$-window cutoff result for a non-monotone finite-range spin system on $\mathbb Z^d$. We further show that, despite the lack of monotonicity, a monochromatic initial configuration is asymptotically near-extremal for the one-site magnetization.
The main obstacle is the loss of monotonicity for $q\geq 3$, which prevents the direct use of classical couplings and FKG-based comparisons that underpin sharp high-temperature results for monotone systems. We overcome this by developing an information-percolation framework adapted to signed influences. A key step is a one-site decomposition of the conditional spin law into a uniform component plus a remainder, enabling a refinement of the classical red/green/blue classification. Combining geometric control of backward histories with Fourier-analytic bounds for signed convolution powers yields a precise identification of the cutoff location and an optimal $O(1)$ window.
Jaeyun Yi (SLMath)
Title: Quantum field theory and stochastic PDEs
Abstract: In this talk, I will introduce the connection between quantum field theory and stochastic PDEs through stochastic quantisation. As an example, I will discuss the construction of the Phi4 model in R^3 and its axiomatic properties. In particular, I will demonstrate the unique ergodicity of the Phi4 measure at high temperature, along with the exponential decay of correlation. This is based on joint work with Pawel Duch, Martin Hairer, and Wenhao Zhao.