中科院随机分析中心概率讨论班

时间: 2025年5月9日（周五） 15：00-16：00
地点: 南楼613
报告人: 夏傲腾（北京大学）
题目: Near-critical behavior of random field Ising model
摘要: We study the two dimensional random field Ising model in a box with size N where the external field is given by i.i.d. normal variables with mean 0 and variance \eps^2 and derive the following phase transition of boundary influence (i.e., the difference between the spin averages at the center of the box with the plus and the minus boundary conditions) at critical temperature T_c=T_c(2): For \eps \ll N^{-7/8}, the boundary influence decays as N^{-1/8}; for \eps \gg N^{-7/8}, the boundary influence decays as N^{-1/8}e^{-\Theta(\eps^{8/7}N)}. We also study the two dimensional random filed FK-Ising model. The total variation (TV) distance between the FK-Ising measures with and without external filed has the following phase transition: The critical order for \eps is N^{-1}, N^{-15/16} and N^{-1/2} for T \gt T_c, T=T_c and 0\lt T \lt T_c respectively; in each case, above this critical order, the TV distance converges to 1 as N goes to infinity and to 0 below. This talk is based on joint works with Jian Ding, Chenxu Hao and Fenglin Huang.

时间: 2025年5月9日（周五） 16：00-17：00
地点: 南楼613
报告人: 宋健（山东大学）
题目: Anticipated backward stochastic evolution equations and maximum principle for path-dependent systems in infinite dimension
摘要: For a class of path-dependent stochastic evolution equations driven by cylindrical $Q$-Wiener process, we study the Pontryagin's maximum principle for the stochastic recursive optimal control problem. In this infinite-dimensional control system, the state process depends on its past trajectory, the control is delayed via an integral with respect to a general finite measure, and the final cost relies on the delayed state. To obtain the maximum principle, we introduce a new type of non-anticipative path derivative and its dual operator, which allows us to derive an anticipated backward stochastic evolution equation as the adjoint equation of the state equation. This is joint work with Guomin Liu and Meng Wang.

时间: 2025年5月23日（周五） 15：00-16：00
地点: 南楼613
报告人: 张朝恩（哈尔滨工业大学）
题目: The logarithmic Sobolev inequality and its nonlinear analogues
摘要: The entropy producing phenomenon is fundamental in physics. It demonstrates the tendency of non-equilibrium states to equilibrium states as a system evolves. In the mathematical study of kinetic theory, it is one of the key goals to describe this phenomenon quantitatively and rigorously. The logarithmic Sobolev inequality and its nonlinear analogues have played an important role in this direction. In this talk, I will present results and questions about this functional inequality approach. I will start by introducing basic facts and some related questions/progresses about logarithmic Sobolev inequality. Then I will speak about its nonlinear analogues for the following models: the McKean-Vlasov equation, the Landau equation and the Kac model.

时间: 2025年5月23日（周五） 16：00-17：00
地点: 南楼613
报告人: 袁望钧（南方科技大学）
题目: Multiple collisions of eigenvalues and singular values of matrix Gaussian field
摘要: Let $X^\beta$ be a real symmetric or complex Hermitian matrix whose entries are independent Gaussian random fields. We provide the sufficient and necessary conditions such that multiple collisions of eigenvalue processes of $A^\beta + T_\beta X^\beta T_\beta^*$ occur with positive probability. In addition, for a real or complex rectangular matrix $W^\beta$ with independent Gaussian random field entries, we obtain the sufficient and necessary conditions under which the probability of multiple collisions of non-trivial singular value processes of $B^\beta + T_\beta W^\beta \tilde T_\beta$ is positive. In both cases, the size of the set of collision times is characterized via Hausdorff dimension.

时间: 2025年6月6日（周五） 15：00-16：00
地点: 南楼613
报告人: 杨孟（大湾区大学）
题目: Planar Orthogonal Polynomials and Their Applications
摘要: Planar orthogonal polynomials play an important role in studying the statistical behavior of the eigenvalues of random normal matrix ensembles. In this talk, I will present the strong asymptotics of planar orthogonal polynomials for the Gaussian weight with logarithmic singularities, and then describe their applications in computing correlation kernels and partition functions. This talk is based on recent works with Seung-Yeop Lee, Torben Kruger, Sung-Soo Byun, and Seong-Mi Seo.

时间: 2025年6月6日（周五） 16：00-17：00
地点: 南楼613
报告人: 吴保君（北京大学）
题目: Random geometry on the annulus
摘要: In 1981, Polyakov introduced the bosonic string measure as the path integral of 2d random surfaces. This work has profoundly motivated recent mathematical advances in Liouville conformal field theory (LCFT) and the study of random geometries. In the first part of the talk, I explain the LCFT on the annulus and derive its partition function. In the second part, I explain some applications to Brownian loop soups (BLS), and we give explicit formulas for some random moduli problems of Brownian loop soups. Which is based on the joint work with Gefei Cai, Jiaqi Liu, Wei Qian and Xin Sun.

时间: 2025年6月20日（周五） 15：00-16：00
地点: 南楼613
报告人: 姚东（江苏师范大学）
题目: Small gap problem of point processes
摘要: In the talk, I will give a brief review of the extreme gap problems (smallest and largest gaps of the eigenvalues) of various random matrix ensembles. I will also present our recent series of work on smallest gaps of several random point processes associated with Gaussian structures.

时间: 2025年6月20日（周五） 16：00-17：00
地点: 南楼613
报告人: 解龙杰（江苏师范大学）
题目: Homogenized limit of the fully coupled multi-scale non-linear stochastic system
摘要: We are concerned with the asymptotic behavior of the fully coupled multi-scale McKean-Vlasov stochastic system involving the joint distribution of the slow and fast motions. By studying the regularities as well as the long time estimates of the solutions of the mean field type Kolmogorov equation and the degenerate Poisson equation in Wasserstein space, we establish quantitative homogenized limit of the whole non-linear system, which seems to be new even for classical multi-scale SDEs. Application to the over-damped limit of kinetic Vlasov-Fokker-Planck equation is also provided.

时间: 2025年7月4日（周五） 15：00-16：00
地点: 南楼613
报告人: 亓维维（中科院）
题目: Existence and regularity of the quasi-potential function in LDP
摘要: Real systems are often subject to noises due to internal uncertainties and complexity as well as external randomness. Even small noises can lead to intriguing dynamics across multiple timescales including transient dynamics and long-term dynamics captured by quasi-stationary distributions and stationary distributions, respectively. Understanding the asymptotic behaviors of (quasi-)stationary distributions in the small noise limit is of fundamental importance. However, this analysis presents a singular limit, posing significant challenges. A powerful approach is to establish the large deviation principle (LDP) for (quasi-)stationary distributions. In this talk, we will present our recent results on the existence and regularity of the quasi-potential function in LDP, which have broad and impactful applications across various disciplines. Specifically, we will discuss its applications in non-equilibrium thermodynamics.

时间: 2025年7月4日（周五） 16：00-17：00
地点: 南楼613
报告人: 刘党政（中国科学技术大学）
题目: Outliers for deformed inhomogeneous random matrices
摘要: Inhomogeneous random matrices with a non-trivial variance profile determined by a symmetric Markov matrix and with independent symmetrically sub-Gaussian entries up to Hermitian symmetry, include many prominent examples, such as Wigner matrices, remarkable sparse Wigner matrices and random band matrices, and have been of great interest recently. In this talk, we study low-rank additive perturbations of such random matrices, and establish a sharp BBP phase transition for extreme eigenvalues at the level of law of large numbers. Under suitable conditions on the variance profile and the finite-rank perturbation, we also establish the fluctuations of spectral outliers that may be characterized by the general inhomogeneous random matrices. This reveals the strong non-universality phenomena, which may depend on eigenvectors, sparsity or geometric structure. Based on joint work with Ruohan Geng and Guangyi Zou.

时间: 2025年7月25日（周五） 15：00-16：00
地点: 南楼613
报告人: 朱庆三（南京航空航天大学）
题目: Critical branching random walks, branching capacity and branching interlacements
摘要: We discuss critical branching random walks and introduce new concepts for critical branching random walks: branching capacity and branching interlacements. By introducing these concepts, we obtain analogues of various classical results for random walks, in the setting of critical branching random walks: the exact asymptotics for the visiting probability, the local limit of branching random walks in tori, the cover time of a torus by branching random walks, etc.

时间: 2025年7月25日（周五） 16：00-17：00
地点: 南楼613
报告人: 廖羽晨（中国科学技术大学）
题目: Space-time joint laws of the KPZ fixed point
摘要: The KPZ fixed point, first constructed by Matetski-Quastel-Remenik, is a universal scaling limit for the so-called Kardar-Parisi-Zhang universality class, a 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. Based on joint work with Zhipeng Liu.