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April 2016 Cutoff for the noisy voter model
J. Theodore Cox, Yuval Peres, Jeffrey E. Steif
Ann. Appl. Probab. 26(2): 917-932 (April 2016). DOI: 10.1214/15-AAP1108

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.

Citation

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J. Theodore Cox. Yuval Peres. Jeffrey E. Steif. "Cutoff for the noisy voter model." Ann. Appl. Probab. 26 (2) 917 - 932, April 2016. https://doi.org/10.1214/15-AAP1108

Information

Received: 1 September 2014; Revised: 1 January 2015; Published: April 2016
First available in Project Euclid: 22 March 2016

zbMATH: 1339.60109
MathSciNet: MR3476629
Digital Object Identifier: 10.1214/15-AAP1108

Subjects:
Primary: 60J27, 60K35

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 2 • April 2016
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