Consider the complete graph on n vertices. To each vertex assign an Ising spin that can take the values -1\documentclass[12pt]{minimal}
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\begin{document}$$-1$$\end{document} or +1\documentclass[12pt]{minimal}
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\begin{document}$$+1$$\end{document}. Each spin i∈[n]={1,2,⋯,n}\documentclass[12pt]{minimal}
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\begin{document}$$i \in [n]=\{1,2,\dots , n\}$$\end{document} interacts with a magnetic field h∈[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$h \in [0,\infty )$$\end{document}, while each pair of spins i,j∈[n]\documentclass[12pt]{minimal}
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\begin{document}$$i,j \in [n]$$\end{document} interact with each other at coupling strength n-1J(i)J(j)\documentclass[12pt]{minimal}
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\begin{document}$$n^{-1} J(i)J(j)$$\end{document}, where J=(J(i))i∈[n]\documentclass[12pt]{minimal}
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\begin{document}$$J=(J(i))_{i \in [n]}$$\end{document} 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,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\beta \in (0,\infty )$$\end{document}. We show that there are critical thresholds βc\documentclass[12pt]{minimal}
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\begin{document}$$\beta _c$$\end{document} and hc(β)\documentclass[12pt]{minimal}
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\begin{document}$$h_c(\beta )$$\end{document} such that, in the limit as n→∞\documentclass[12pt]{minimal}
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\begin{document}$$n\rightarrow \infty $$\end{document}, the system exhibits metastable behaviour if and only if β∈(βc,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\beta \in (\beta _c, \infty )$$\end{document} and h∈[0,hc(β))\documentclass[12pt]{minimal}
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\begin{document}$$h \in [0,h_c(\beta ))$$\end{document}. Our main result is a sharp asymptotics, up to a multiplicative error 1+on(1)\documentclass[12pt]{minimal}
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\begin{document}$$1+o_n(1)$$\end{document}, 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\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{n}$$\end{document} times a centred Gaussian random variable with a complicated variance depending on β,h\documentclass[12pt]{minimal}
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\begin{document}$$\beta ,h$$\end{document}, on the law of J and on the metastable state. The critical thresholds βc\documentclass[12pt]{minimal}
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\begin{document}$$\beta _c$$\end{document} and hc(β)\documentclass[12pt]{minimal}
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\begin{document}$$h_c(\beta )$$\end{document} depend on the law of J, and so does the number of metastable states. We derive an explicit formula for βc\documentclass[12pt]{minimal}
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\begin{document}$$\beta _c$$\end{document} and identify some properties of β↦hc(β)\documentclass[12pt]{minimal}
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\begin{document}$$\beta \mapsto h_c(\beta )$$\end{document}. Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.
