Mixing and cut-off in cycle walks

Abstract

Given a sequence $(\mathfrak{X} _i, \mathscr{K} _i)_{i=1}^\infty $ of Markov chains, the cut-off phenomenon describes a period of transition to stationarity which is asymptotically lower order than the mixing time. We study mixing times and the cut-off phenomenon in the total variation metric in the case of random walk on the groups $\mathbb{Z} /p\mathbb{Z} $, $p$ prime, with driving measure uniform on a symmetric generating set $A \subset \mathbb{Z} /p\mathbb{Z} $.