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.