Random walks on finite groups converging after finite number of steps

• Published in: Algebra and Discrete Mathematics
• Date: 2008
• Main Authors: Vyshnevetskiy, A.L., Zhmud, E.M.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2008
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/153370
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Random walks on finite groups converging after finite number of steps / A.L. Vyshnevetskiy, E.M. Zhmud // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 2. — С. 123–129. — Бібліогр.: 3 назв. — англ.

Let P be a probability on a finite group G, P(n)=P∗…∗P (n times) be an n-fold convolution of P. If n→∞, then under mild conditions P(n) converges to the uniform probability U(g)=1|G| (g∈G). We study the case when the sequence P(n) reaches its limit U after finite number of steps: P(k)=P(k+1)=⋯=U for some k. Let Ω(G) be a set of the probabilities satisfying to that condition. Obviously, U∈Ω(G). We prove that Ω(G)≠U for ``almost all'' non-Abelian groups and describe the groups for which Ω(G)=U. If P∈Ω(G), then P(b)=U, where b is the maximal degree of irreducible complex representations of the group G.