Random walks on finite groups converging after finite number of steps
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...
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| Дата: | 2018 |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/814 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-8142018-03-22T09:39:19Z Random walks on finite groups converging after finite number of steps Vyshnevetskiy, A. L. Zhmud, E. M. random walks on groups, finite groups, group algebra 20P05, 60B15 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 \(G\). Lugansk National Taras Shevchenko University 2018-03-22 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/814 Algebra and Discrete Mathematics; Vol 7, No 2 (2008) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/814/344 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
random walks on groups finite groups group algebra 20P05 60B15 |
| spellingShingle |
random walks on groups finite groups group algebra 20P05 60B15 Vyshnevetskiy, A. L. Zhmud, E. M. Random walks on finite groups converging after finite number of steps |
| topic_facet |
random walks on groups finite groups group algebra 20P05 60B15 |
| format |
Article |
| author |
Vyshnevetskiy, A. L. Zhmud, E. M. |
| author_facet |
Vyshnevetskiy, A. L. Zhmud, E. M. |
| author_sort |
Vyshnevetskiy, A. L. |
| title |
Random walks on finite groups converging after finite number of steps |
| title_short |
Random walks on finite groups converging after finite number of steps |
| title_full |
Random walks on finite groups converging after finite number of steps |
| title_fullStr |
Random walks on finite groups converging after finite number of steps |
| title_full_unstemmed |
Random walks on finite groups converging after finite number of steps |
| title_sort |
random walks on finite groups converging after finite number of steps |
| description |
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 \(G\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/814 |
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AT vyshnevetskiyal randomwalksonfinitegroupsconvergingafterfinitenumberofsteps AT zhmudem randomwalksonfinitegroupsconvergingafterfinitenumberofsteps |
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2024-04-12T06:25:51Z |
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2024-04-12T06:25:51Z |
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