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Algebra and Discrete Mathematics, 2008, выпуск 2, страницы 123–129
(Mi adm164)
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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
RESEARCH ARTICLE
Random walks on finite groups converging after finite number of steps
A. L. Vyshnevetskiya, E. M. Zhmud' a Karazina st. 7/9, apt. 34, 61078, Kharkov,
Ukraine
Аннотация:
Let $P$ be a probability on a finite group $G$, $P^{(n)}=P\ast\ldots\ast P$ ($n$ times) be an $n$-fold convolution of $P$. If $n\rightarrow\infty$, then under mild conditions $P^{(n)}$ converges to the uniform probability $U(g)=\frac 1{|G|}$ $(g\in G)$. We study the case when the sequence $P^{(n)}$ reaches its limit $U$ after finite number of steps: $P^{(k)}=P^{(k+1)}=\dots=U$ for some $k$. Let $\Omega(G)$ be a set of the probabilities satisfying to that condition. Obviously, $U\in\Omega(G)$. We prove that $\Omega(G)\neq U$ for “almost all” non-Abelian groups and describe the groups for which $\Omega(G)=U$. If $P\in \Omega(G)$, then $P^{(b)}=U$, where $b$ is the maximal degree of irreducible complex representations of the group $G$.
Ключевые слова:
random walks on groups, finite groups, group algebra.
Образец цитирования:
A. L. Vyshnevetskiy, E. M. Zhmud', “Random walks on finite groups converging after finite number of steps”, Algebra Discrete Math., 2008, no. 2, 123–129
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/adm164 https://www.mathnet.ru/rus/adm/y2008/i2/p123
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