Random walks on finite groups converging after finite number of steps

Abstract

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 \(G\).