Markov chain mixting time, card shuffling and spin systems dynamics

Abstract

The mixing time of a Markov chain describes how fast the Markov chain converges to its stationary distribution. In this thesis, we survey some of the knowledge and main tools available in this field by looking at examples. We focus on various models of card shuffling (random walk on the permutation group $S_n$) and the Swendsen-Wang dynamics of the mean-field Ising Model. We show that the Card-Cyclic to Random shuffle has mixing time of order $\Theta(n \log n)$ (joint work of Ben Morris and Yuval Peres). We also determine the order of the mixing time of the mean field Swendsen-Wang dynamics at all temperatures. In particular, at criticality, it mixes at time $\Theta(n^{1\over 4})$ (joint work of Yun Long, Asaf Nachmias, and Yuval Peres).