2019/12/23 by Frank den Hollander, Hollander, Frank den, Oliver Jovanovski +1 · 2 citations
Physics and Astronomy · Mathematics · #Opinion Dynamics and Social Influence #advanced mathematical theories #Complex Network Analysis Techniques
paper · pdf · doi:10.48550/arxiv.1912.10591
We investigate the effect of disorder on the Curie-Weiss model with Glauber dynamics. In particular, we study metastability for spin-flip dynamics on the Erdős-Rényi random graph ERn(p) with n vertices and with edge retention probability p ∈ (0,1). Each vertex carries an Ising spin that can take the values -1 or +1. Single spins interact with an external magnetic field h ∈ (0,∞), while pairs of spins at vertices connected by an edge interact with each other with ferromagnetic interaction strength 1/n. Spins flip according to a Metropolis dynamics at inverse temperature β. The standard Curie-Weiss model corresponds to the case p=1, because ERn(1) = Kn is the complete graph on n vertices. For β>βc and h ∈ (0,p χ(βp)) the system exhibits metastable behaviour in the limit as n→∞, where βc=1/p is the critical inverse temperature and χ is a certain threshold function satisfying limλ→∞ χ(λ) =1 and limλ\downarrow 1 χ(λ)=0. We compute the average crossover time from the metastable set (with magnetization corresponding to the `minus-phase') to the stable set (with magnetization corresponding to the `plus-phase'). We show that the average crossover time grows exponentially fast with n, with an exponent that is the same as for the Curie-Weiss model with external magnetic field h and with ferromagnetic interaction strength p/n. We show that the correction term to the exponential asymptotics is a multiplicative error term that is at most polynomial in n. For the complete graph Kn the correction term is known to be a multiplicative constant.