Almost invariance of distributions for random walks on groups

• 成果类型: Article
• 署名作者: Erschler, Anna
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00915-3
• 发表日期: 2019
• 页码: 445-476
• 关键词: growth

摘要

We study the neighborhoods of a typical point Zn visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to n(Zn). If such neighborhood contains a ball of radius Cn, we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of Z with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa's criterion for the property HFD. We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n/L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of Z2 with a finite group is asymptotically strictly larger than n/L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.