Metastability for Glauber dynamics on the complete graph with coupling disorder

Abstract

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 $\beta \in (0,\infty )$. We show that there are critical thresholds ${\beta }_c$ and $h_c(\beta )$ such that, in the limit as $n\to \infty$, the system exhibits metastable behaviour if and only if $\beta \in ({\beta }_c,\infty )$ and $h\in [0,h_c(\beta ))$. 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 $\beta ,h$, on the law of $J$ and on the metastable state. The critical thresholds ${\beta }_c$ and $h_c(\beta )$ depend on the law of $J$, 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.

41 pages, 6 figures

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