Cutoff for the noisy voter model

Abstract

Given a continuous time Markov Chain $\{q(x,y)\}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on $\{0,1\}^{S}$, which evolves in the following way: (1) for each two sites $x$ and $y$ in $S$, the state at site $x$ changes to the value of the state at site $y$ at rate $q(x,y)$; (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates $\{q(x,y)\}$ and the corresponding stationary distributions are almost uniform, then the mixing time has a sharp cutoff at time $\log|S|/2$ with a window of order 1. Lubetzky and Sly proved cutoff with a window of order 1 for the stochastic Ising model on toroids; we obtain the special case of their result for the cycle as a consequence of our result. Finally, we consider the model on a star and demonstrate the surprising phenomenon that the time it takes for the chain started at all ones to become close in total variation to the chain started at all zeros is of smaller order than the mixing time.