## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 26, Number 2 (2016), 917-932.

### Cutoff for the noisy voter model

J. Theodore Cox, Yuval Peres, and Jeffrey E. Steif

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 26, Number 2 (2016), 917-932.

**Dates**

Received: September 2014

Revised: January 2015

First available in Project Euclid: 22 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1458651824

**Digital Object Identifier**

doi:10.1214/15-AAP1108

**Mathematical Reviews number (MathSciNet)**

MR3476629

**Zentralblatt MATH identifier**

1339.60109

**Subjects**

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Noisy voter models mixing times for Markov chains cutoff phenomena

#### Citation

Cox, J. Theodore; Peres, Yuval; Steif, Jeffrey E. Cutoff for the noisy voter model. Ann. Appl. Probab. 26 (2016), no. 2, 917--932. doi:10.1214/15-AAP1108. https://projecteuclid.org/euclid.aoap/1458651824