Asymptomatic behaviour of the noisy voter model density process

Pymar, Richard and Rivera, N. (2024) Asymptomatic behaviour of the noisy voter model density process. The Annals of Applied Probability 34 (5), pp. 4554-4594. ISSN 1050-5164.

Abstract

Given a transition matrix P indexed by a finite set V of vertices, the voter model is a discrete-time Markov chain in { 0 , 1 } V where at each time-step a randomly chosen vertex x imitates the opinion of vertex y with probability P ( x , y ) . The noisy voter model is a variation of the voter model in which vertices may change their opinions by the action of an external noise. The strength of this noise is measured by an extra parameter p ∈ [ 0 , 1 ] . In this work we analyse the density process, defined as the stationary mass of vertices with opinion 1, that is, S t = ∑ x ∈ V π ( x ) ξ t ( x ) , where π is the stationary distribution of P, and ξ t ( x ) is the opinion of vertex x at time t. We investigate the asymptotic behaviour of S t when t tends to infinity for different values of the noise parameter p. In particular, by allowing P and p to be functions of the size | V | , we show that, under appropriate conditions and small enough p a normalised version of S t converges to a Gaussian random variable, while for large enough p, S t converges to a Bernoulli random variable. We provide further analysis of the noisy voter model on a variety of specific graphs including the complete graph, cycle, torus, and hypercube, where we identify the critical rate p (depending on the size | V | ) that separates these two asymptotic behaviours.

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