Asymptomatic behaviour of the noisy voter model density process

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.