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 $i in [n]={1,2,dots, n}$ interacts with a magnetic field $h in [0,infty)$, while each pair of spins $i,j in [n]$ interact with each other at coupling strength $n^{-1} J(i)J(j)$, where $J=(J(i))_{i in [n]}$ are i.i.d. non-negative random variables drawn from a prescribed probability distribution $mathcal{P}$. Spins flip according to a Metropolis dynamics at inverse temperature $beta in (0,infty)$. We show that there are critical thresholds $beta_c$ and $h_c(beta)$ such that, in the limit as $ntoinfty$, the system exhibits metastable behaviour if and only if $beta in (beta_c, infty)$ and $h in [0,h_c(beta))$. Our main result are 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 $sqrt{n}$ times a centred Gaussian random variable with a complicated variance depending on $beta,h,mathcal{P}$ and on the metastable state. The critical thresholds $beta_c$ and $h_c(beta)$ depend on $mathcal{P}$, and so does the number of metastable states. We derive an explicit formula for $beta_c$ and identify some properties of $beta mapsto h_c(beta)$. Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.