Metastability for Glauber Dynamics on the Complete Graph with Coupling Disorder

Consider the complete graph on n vertices. To each vertex assign an Ising spin that can take the values -1 or +1 . Each spin i ∈ [n]={1,2,… , n} interacts with a magnetic field h ∈ [0,∞ ) , while each pair of spins i,j ∈ [n] interact with each other at coupling strength n^-1 J(i)J(j) , where J=(J(i))_i ∈ [n] are i.i.d. non-negative random variables drawn from a probability distribution with finite support. Spins flip according to a Metropolis dynamics at inverse temperature β∈ (0,∞ ) . We show that there are critical thresholds β _c and h_c(β ) such that, in the limit as n→∞ , the system exhibits metastable behaviour if and only if β∈ (β _c, ∞ ) and h ∈ [0,h_c(β )) . Our main result is a sharp asymptotics, up to a multiplicative error 1+o_n(1) , of the average crossover time from any metastable state to the set of states with lower free energy. We use standard techniques of the potential-theoretic approach to metastability. The leading order term in the asymptotics does not depend on the realisation of J , while the correction terms do. The leading order of the correction term is √(n) times a centred Gaussian random variable with a complicated variance depending on β ,h , on the law of J and on the metastable state. The critical thresholds β _c and h_c(β ) depend on the law of J , and so does the number of metastable states. We derive an explicit formula for β _c and identify some properties of β↦ h_c(β ) . Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.