Conditions of convergence of a random walk on a finite group

A. L. Vyshnevetskiy Colloquium Mathematicum MSC: Primary 60B15, 60B10; Secondary 20D99. DOI: 10.4064/cm8196-5-2020 Published online: 12 May 2021

Abstract

Let $P$ be a probability on a finite group $G$, and $V$ its carrier. Conditions of convergence of the $n$-fold convolution $P^{(n)}$ to the uniform probability on $G$ ($n\rightarrow \infty $) in terms of $V$ are known. We consider conditions under which the carrier $V^{n}$ of the probability $P^{(n)}$ converges as $n\rightarrow \infty $, i.e. $V^{n}=V^ {n+1}=\cdots $ for $n$ large. The convergence of $V^{n}$ is equivalent to the convergence of $P^{(n)}$. Instead of $G$ one can take its subgroup $\langle V \rangle $, generated by $V$. Then $V^{n}$ does not converge if and only if $V$ lies in a non-identity coset in $\langle V \rangle $ of a normal subgroup with a cyclic factor group. The article can be considered as a study of the behavior of the powers $V^{n}$ of a subset $V$ of a finite group as $n\rightarrow \infty $.

Authors

  • A. L. VyshnevetskiyKharkiv National Automobile and Highway University
    Yaroslava Mudrogo St. 25
    61002, Kharkiv, Ukraine
    e-mail

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