ASYMPTOTIC BEHAVIOUR OF THE NOISY VOTER MODEL DENSITY PROCESS

• 成果类型: Article
• 署名作者: Pymar, Richard; Rivera, Nicolas
• 署名单位: University of London; University of Greenwich; Universidad de Valparaiso
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2074
• 发表日期: 2024
• 页码: 4554-4594
• 关键词: steins method; coalescence time; random-variables; random-walks

摘要

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 is an element of [0, 1]. In this work we analyse the density process, defined as the stationary mass of vertices with opinion 1, that is, St = x is an element of V pi (x) t (x), where pi is the stationary distribution of P, and t(x) is the opinion of vertex x at time t. We investigate the asymptotic behaviour of St 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 St converges to a Gaussian random variable, while for large enough p, St 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.