First hitting times for some random walks on finite groups (Q1303910)

Let \((W_n)_{n\geq 0}\) be a random walk on a finite group \(G\) defined by \(W_n=W_0\cdot X_1\cdot \ldots \cdot X_n\), \(n > 0\), where \(X_0\) is uniformly distributed on \(G\) and the \(X_i\) are i.i.d. with a distribution concentrated on a generating subset \(E\) of \(G\) being the union of conjugacy classes. The author studies the distribution of the hitting time \(T=\inf \{ n : W_n=e\}\) for the identity element \(e\) of \(G\). Given conditions on the group (rather on its irreducible representations) it is shown that for large groups \(G\) the distribution of \(T\) is close to the exponential distribution. An upper bound for the (Kolmogorov-)distance is given. The result is applied to special examples such as the symmetric groups and some classical groups.