Asymptotic 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 $p \in [0,1]$. In this work we analyse the density process, defined as the stationary mass of vertices with opinion 1, i.e. $S_t = \sum_{x\in V} \pi(x)\xi_t(x)$, where $\pi$ is the stationary distribution of $P$, and $\xi_t(x)$ 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.