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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 probability distribution with finite support. Spins flip according to a Metropolis dynamics at inverse temperature $β\in (0,\infty)$. We show that there are critical thresholds $β_c$ and $h_c(β)$ such that, in the limit as $n\to\infty$, the system exhibits metastable behaviour if and only if $β\in (β_c, \infty)$ and $h \in [0,h_c(β))$. 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 $\sqrt{n}$ 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 $β\mapsto h_c(β)$. Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.