中科院随机分析中心概率讨论班

时间: 2025年9月12日（周五） 15：00-16：00
地点: 南楼613
报告人: 蒲飞（北师大）
题目: Spatial decorrelation of KPZ from narrow wedge
摘要: In this talk, I will present the spatial decorrelation of the solution to the KPZ equation with narrow wedge initial data. For fixed $t>0$, we determine the decay rate of the spatial covariance function, showing that $\Cov[h(t,x),h(t,0)]\sim \frac{t}{x}$ as $x\to\infty$. In addition, we prove that the finite-dimensional distributions of the properly rescaled spatial average of the height function converge to those of a Brownian motion. This is based on a joint work with Yu Gu.

时间: 2025年9月12日（周五） 16：00-17：00
地点: 南楼613
报告人: 侯浩杰（北理工）
题目: On the maximal displacement of subcritical branching random walks with or without killing
摘要: Consider a subcritical branching random walk $\{Z_k\}_{k\geq 0}$ with offspring distribution $\{p_k\}_{k\geq 0}$ and step size $X$. Let $M_n$ denote the rightmost position reached by $\{Z_k\}_{k\geq 0}$ up to generation $n$ and define $M := \sup_{n\geq 0} M_n$. In this talk, we give asymptotics of tail probability of $M$ under optimal assumptions $\sum^{\infty}_{k=1}(k\log k) p_k<\infty$ and $\mathbb{E}[Xe^{\gamma X}]<\infty$, where $\gamma >0$ is a constant such that $\mathbb{E}[e^{\gamma X}]=\frac{1}{m}$ and $m=\sum_{k=0}^\infty kp_k\in (0,1)$. Moreover, we confirm the conjecture of Neuman and Zheng [Probab. Theory Related Fields, 2017] by establishing the existence of a critical value $m\mathbb{E}[X e^{\gamma X}]$ such that \begin{align} \lim_{n\to\infty} \mathbb{P}(M_n\geq cn\big| M\geq cn)= \begin{cases} &1,~c\in\big(0,m\mathbb{E}[Xe^{\gamma X}]\big); \\ &0,~c\in\big(m\mathbb{E}[Xe^{\gamma X}], \infty\big). \end{cases} \end{align} Finally, we extend these results to the maximal displacement of branching random walks with killing. Based on a joint work with Shuxiong Zhang (Anhui Normal University).

时间: 2025年9月19日（周五） 14：00-15：00
地点: 南楼613
报告人: 苏中根（浙江大学）
题目: Couplings, Strong Invariance Principles with Applications to Random Walks on Inifinite Percolation Clusters
摘要: This talk basically consists of two parts. First, I will briefly review some classical coupling methods and strong invariance principles for sums of independent random variables in probability theory. Then I will report a recent work on couplings between Brownian motions and random walks on the infinite cluster in the supercritical Bernoulli bond percolation on $\mathbb{Z}^d$. As a corollary, we recover the law of the iterated logarithm proved by Duminil-Copin (arXiv:0809.4380) and further identify the limit constant. This talk is mainly based on a joint work with C.Gu and R. Xu (PTRF, 2025).

时间: 2025年9月24日（周三） 15：00-16：00
地点: 南楼205
报告人: 秦硕（BIMSA）
题目: Step Reinforced Random Walks in Euclidean Space and on Finite Groups
摘要: At each time step, with a given probability, the step-reinforced random walk randomly selects one of its previous steps and repeats that step; otherwise it takes a fresh step sampled from a fixed distribution. In Euclidean space, we show that the walk exhibits a recurrence/transience phase transition in dimensions one and two, whereas it remains transient for all parameters in dimensions three and above. We will also report recent progress regarding mixing times of step-reinforced random walks on finite groups.

时间: 2025年9月26日（周五） 15：00-16：00
地点: 南楼613
报告人: 王振富（北京大学）
题目: Kac’s program for the Landau equation
摘要: We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative N -particle system, obtained by passing to the grazing limit on Kac’s walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials, including the physically important Coulomb case. This provides the first resolution of propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions, and the first extension of Kac’s program to the Landau equation in the soft potential regime. The convergence is established in weak, Wasserstein, and entropic senses,together with strong L1 convergence. To handle the singularity of soft potentials, we extend the duality approach of Bresch-Duerinckx-Jabin and establish key functional inequalities, including an extended commutator estimate and a new second-order Fisher information estimate. Based on a joint work with Xuanrui Feng (PKU).

时间: 2025年9月26日（周五） 16：00-17：00
地点: 南楼613
报告人: 王雄（中山大学）
题目: Interacting Particle Systems on Networks
摘要: Modeling multi-agent systems on networks is a fundamental challenge in a wide variety of disciplines. We jointly infer the weight matrix of the network and the interaction kernel, which determine respectively which agents interact with which others and the rules of such interactions from data consisting of multiple trajectories. The estimator we propose leads naturally to a non-convex optimization problem, and we investigate two approaches for its solution: one is based on the alternating least squares (ALS) algorithm; another is based on a new algorithm named operator regression with alternating least squares (ORALS). Both algorithms are scalable to large ensembles of data trajectories. We establish coercivity conditions guaranteeing identifiability and well-posedness. The ALS algorithm appears statistically efficient and robust even in the small data regime but lacks performance and convergence guarantees. The ORALS estimator is consistent and asymptotically normal under a coercivity condition. We conduct several numerical experiments ranging from Kuramoto particle systems on networks to opinion dynamics in leader-follower models.

时间: 2025年10月10日（周五） 15：00-16：00
地点: 南楼613
报告人: 刘明昶（首都师范大学）
题目: $\beta$-ensemble in the scaling limit of discrete models
摘要: The $\beta$-ensemble is a mathematical model describing the distribution of charged particles with repulsive interactions, where the parameter $\beta>0$ controls the interaction strength. Schramm- Loewner Evolution (SLE) is a family of curves parameterised by $\kappa>0$ describing the scaling limits of discrete statistical models at criticality. It has been conjectured that $\beta$-ensemble and SLE$_\kappa$ curve are connected when $\beta=\frac{2}{\kappa}$ and $\beta=\frac{8}{\kappa}$. A known example is that the driving function of certain multiple SLEs, when parameterised by common parameterisation, coincides with Dyson’s Brownian motion—a dynamic version of $\beta$-ensemble. However, a direct connection between $\beta$-ensemble itself and SLE$_\kappa$ remains unclear. In this report, I will discuss two discrete models: loop-erased random walk and Ising model. Through these examples, I will explain that how $\beta$-ensembles describe the distribution of hitting points of certain multiple SLE$_\kappa$ curves. Inspired by this relation, I will also explain how to derive the scaling limit of certain connection probabilities in these two discrete models.

时间: 2025年10月10日（周五） 16：00-17：00
地点: 南楼613
报告人: 李韫（清华大学）
题目: Limits of the truncated circular beta ensembles
摘要: Consider a Haar unitary matrix with the first row and column deleted, Zyczkowski and Sommers derived the joint distribution of the eigenvalues, and showed that they form a determinantal point process. Killip and Kozhan extended this result to circular beta ensembles, and provided a description of the spectrum of the truncated version of the circular beta ensembles (with beta=2 corresponding to the Haar unitary case). In this talk, I will discuss the edge and bulk point process limits of the truncated circular beta ensembles, along with the scaling limits of the normalized characteristic polynomials. The limiting objects are closely connected to the stochastic zeta function and the iid Gaussian power series in the edge and bulk regimes, respectively. I will also explain how the random Dirac-type operator framework can be used to derive scaling limits for the full and truncated circular ensembles. Based on joint works with Mingchang Liu, Joseph Najnudel, and Benedek Valko.

时间: 2025年10月24日（周五） 15：00-16：00
地点: 南楼613
报告人: 黄逸超（北京理工大学）
题目: Scaling limit of the $H^{2|2}$ model on the hierarchical lattice
摘要: It is known since the work of Sabot-Tarrès that the $H^{2|2}$ supersymmetric hyperbolic sigma model is connected to the so-called reinforced Markov processes. I will give a brief overview of some of the recent progress on the vertex reinforced jump process on the Dyson hierarchical lattice. Especially, I will review its connection to a special family of random Schrödinger operator, and describe some exact renormalization procedures that one can perform on the hierarchical lattice. Using these information one can construct probabilistically the scaling limit of the $H^{2|2}$ model on the hierarchical lattice, and we will discuss some basic properties of this random limit. This is an ongoing joint work with Jinglin Wang and Xiaoling Zeng (Strasbourg).

时间: 2025年10月24日（周五） 16：00-17：00
地点: 南楼613
报告人: 向圣权（北京大学）
题目: Exponential mixing for the randomly forced NLS equation
摘要: We explorer exponential mixing of the invariant mesure for randomly forced nonlinear Schrödinger equation, with damping and random noise localized in space. This study emphasizes the crucial role or exponential asymptotic compactness and control properties. This talk is based on the recent joint work with Yuxuan Chen, Zhifei Zhang and Jia-Cheng Zhao.

时间: 2025年11月7日（周五） 14：00-15：00
地点: 南楼613
报告人: 黄璐静（福建师范大学）
题目: The effective resistance in one-dimensional critical long-range percolation
摘要: We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}\d u\d v\}$ for $|i-j|>1$ for some fixed $\beta>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 0 to $[-n,n]^c$ and from the interval $[-n,n]$ to $[-2n,2n]^c$ (conditioned on no edge joining $[-n,n]$ and $[-2n,2n]^c$) both have a polynomial lower bound in n. Our bound holds for all $\beta>0$ and thus rules out a potential phase transition (around $\beta=1$) which seemed to be a reasonable possibility. Finally, I will introduce some other new progresses in this topic. This talk is based on joint works with Jian Ding and Zherui Fan.

时间: 2025年11月7日（周五） 15：00-16：00
地点: 南楼613
报告人: 范哲睿（北京大学）
题目: The effective resistance and random walk in one-dimensional critical long-range percolation
摘要: We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}\d u\d v\}$ for $|i-j|>1$ for some fixed $\beta>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 0 to $[-n,n]^c$ and from the interval $[-n,n]$ to $[-2n,2n]^c$ (conditioned on no edge joining $[-n,n]$ and $[-2n,2n]^c$) both grow like $n^{\delta(\beta)}$ for some $\delta(\beta)\in (0,1)$. Finally, we will consider the heat kernel estimates of the random walk on this model. The talk is based on joint works with Jian Ding and Lu-Jing Huang.

时间: 2025年11月21日（周五） 15：00-16：00
地点: 南楼613
报告人: 赵晓玉（天津大学）
题目: Bismut Formula and Gradient Estimates for Dirichlet Semigroups with Application to Singular Killed DDSDEs
摘要: In this talk, by establishing a local version of Bismut formula for Dirichlet semigroups on a regular domain, gradient estimates are derived for killed SDEs with singular drifts. As an application, the total variation distance between two solutions of killed DDSDEs is bounded above by the truncated $1$-Wasserstein distance of initial distributions, in the regular and singular cases respectively. This work is joint with Professor Feng-Yu Wang.

时间: 2025年11月21日（周五） 16：00-17：00
地点: 南楼613
报告人: 牟宸辰（香港城市大学）
题目: On Well-posedness of Mean Field Game Master Equations: A Unified Approach
摘要: It is well known that the global (in time) well-posedness of mean field game master equations relies on certain monotonicity conditions, and there have been several types of conditions proposed in the literature. In this talk we intend to provide a unified understanding on the role of monotonicity conditions in the theory. Inspired by Lyapunov functions for dynamical systems, we propose a general type of monotonicity condition, which covers all the existing ones as special cases and is essentially necessary for the existence of Lipschitz continuous classical solutions. Our approach works for very general mean field games, including extended mean field games and mean field games with volatility control. In particular, for the latter a new notion of second order monotonicity condition is required. The talk is based on some ongoing joint works with Jianfeng Zhang and Jianjun Zhou.

时间: 2025年12月5日（周五） 15：00-16：00
地点: 南楼613
报告人: 俞锦炯（华东师范大学）
题目: Branching and coalescing random walks
摘要: In this talk, I will review the path properties of coalescing and branching–coalescing random walks in both discrete- and continuous-time settings. I will also discuss related questions arising from their dual voter models and from the universality class of the Brownian web and net. Based on joint work with Rongfeng Sun and Jan M. Swart.

时间: 2025年12月5日（周五） 16：00-17：00
地点: 南楼613
报告人: 刘群（闽南师范大学）
题目: The Sherrington–Kirkpatrick model with vector spins
摘要: The two-dimensional Guerra–Talagrand (GT) bound occupies a pivotal position in the analysis of the Sherrington–Kirkpatrick (SK) model, as exemplified by its application in establishing Talagrand’s positivity of the overlap and disorder chaos. By scrutinizing the Parisi functional introduced by Chen H.-B [arXiv:2311.10446v2], this talk aims to demonstrate the corresponding GT bound for the SK model with vector spins, formulated through optimal stochastic control problems. These results subsequently lead to novel insights regarding Talagrand’s positivity of the overlap matrix in the SK model with vector spins.

时间: 2025年12月12日（周五） 15：00-16：00
地点: 南楼620
报告人: 孙振尧（北理工）
题目: On the subcritical self-catalytic branching Brownian motions
摘要: The self-catalytic branching Brownian motions (SBBM) are extensions of the classical one-dimensional branching Brownian motions by incorporating pairwise branchings catalyzed by the intersection local times of the particle pairs. These processes naturally arise as the moment duals of certain reaction-diffusion equations perturbed by multiplicative space-time white noise. For the subcritical case of the catalytic branching mechanism, we construct the SBBM allowing an infinite number of initial particles. Additionally, we establish the coming down from infinity (CDI) property for these systems and characterize their CDI rates. This is based on a joint work with Haojie Hou.

时间: 2025年12月19日（周五） 15：00-16：00
地点: 南楼613
报告人: 方笑（香港中文大学）
题目: TBA
摘要: TBA

时间: 2025年12月19日（周五） 16：00-17：00
地点: 南楼613
报告人: 蔡志军（南方科技大学）
题目: TBA
摘要: TBA