随机游动研讨会

时间：2024.03.10--03.16

召集人：Yuval   Peres、施展、顾陈琳、石权

Workshop on Random Walks

TianYuan Mathematics Research Center, Yunnan, China

2024/03/10 – 2024/03/16

Sunday 2024/03/10:  Arrival

Saturday 2024/03/16: Departure

Titles and Abstracts

w Chen, Xinxin 陈昕昕 (Beijing Normal University)

Tails of martingale limits in branching random walk

Abstract: We consider a supercritical branching random walk on the real line in the so called case where the whole system a.s. goes to +1 eventually, and the additive martingale converges a.s. and in mean to some non-degenerate random variable W1 under suitable moment condition. We consider the joint tail of the global minimum and W1, and with the help of it, we study the branching random walk conditioned on atypically small global minimum or conditioned on large W1. We will also study the biased random walk in random environment which is given by this branching random walk and talk about some related results. This is based on a joint work with L. de Raphelis.

w Chen, Xinxing 陈新兴 (SJTU)

Some Properties and Open Question for the Derrida-Retaux system

Abstract: We consider a simple max-type recursive model which was introduced in the study of depinning transition in presence of strong disorder, by Derrida and Retaux. Oue interest is focused on the critical regime or nearly critical regime, for which we study the extinction probability and the free energy.

w Fan, Aihua 范爱华 (Picardie University, France/武汉大学)

Some suggestions for research on Khitchin sequences

Abstract: Khintchin conjectured (1923) that for almost every real number x, the sequence {nx mod 1} is strongly  equi-distributed on the interval [0,1]. The conjecture was refuted by Marstrand (1970). However {2^n x mod 1} is strongly equi-distributed  for almost all x. Then rises the question for which   sequences of integers  {a_n}, {a_n x mod 1} is  equi-distributed for almost every x. We know few  results. We are going to suggest problems to study. One of question is how to construct random sequence {a_n(\omega)} such that almost surely {a_n(\omega)x mod 1}

w Gu, Chenlin 顾陈琳 (Tsinghua University)

TBA

Abstract: TBA

w Hu, Zechun 胡泽春 (Sichuan University)

Favorite edges and rarely visited edges of the simple random walks

Abstract: In this talk, we will introduce our recent results on favorite edges and rarely visited edges (i.e., edges that are visited only once) of the one-dimensional simple symmetric random walk based on the following two papers:

[1] C.-X. Hao, Z.-C. Hu, T. Ma, R. Song: Three favorite edges occurs infinitely often for one-dimensional simple random walk, Accepted by Communications in Mathematics and Statistics, 2023.

[2] Z.-C. Hu, X. Peng, R. Song, Y. Tan: The asymptotic behavior of rarely visited edges of the simple random walk, arXiv: 2310.16657v1 (2023).

w Huang, Xiangyu 黄翔宇 (Tsinghua University)

An Introduction to Once-reinforced Random Walks

Abstract: Reinforced random walks are specific non-Markov processes moving on graphs. The original version of this model was introduced by Coppersmith and Diaconis in 1987. Different from the simple random walk, the transition probabilities of reinforced random walks depend on the whole past trajectory. As a specific version of the reinforced random walks, once-reinforced random walks (ORRWs) has a simple reinforcement rule, which in fact leads a further loss of the Markov property on this model. The study on ORRWs is challenging and has no unified method until now. In this talk, I would like to introduce ORRWs and some results on this model..

w Peres, Yuval (BIMSA)

TBA

Abstract: TBA

w Qin, Shuo 秦硕 (NYU Shanghai)

Recurrence and Transience of Multidimensional Elephant Random Walks

Abstract: Abstract: We prove a conjecture by Bertoin that the multidimensional elephant random walk (MERW) on the d-dimensional lattice is transient if d ≥ 3. In dimensions d=1, 2, we prove that phase transitions between recurrence and transience occur at

p=(2 d+1) /(4 d).