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...
Збережено в:
| Дата: | 2018 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/814 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| Резюме: | 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 \(G\). |
|---|