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 values $$-1$$ - 1 or $$+1$$ + . Each $$i \in [n]=\{1,2,\dots , n\}$$ i ∈ [ n ] = { , 2 ⋯ } interacts with a magnetic field $$h [0,\infty )$$ h 0 ∞ ) while pair of spins $$i,j [n]$$ j interact other at coupling strength $$n^{-1} J(i)J(j)$$ J ( where $$J=(J(i))_{i [n]}$$ are i.i.d. non-negative random variables drawn from probability distribution finite support. Spins flip according to Metropolis dynamics inverse temperature $$\beta (0,\infty β We show there critical thresholds _c$$ c and $$h_c(\beta such that, in limit as $$n\rightarrow \infty $$ → system exhibits metastable behaviour if only (\beta _c, [0,h_c(\beta ))$$ Our main result is sharp asymptotics, up multiplicative error $$1+o_n(1)$$ o average crossover time any state set states lower free energy. use standard techniques potential-theoretic approach metastability. The leading order term asymptotics does not depend realisation J correction terms do. $$\sqrt{n}$$ times centred Gaussian variable complicated variance depending ,h$$ law state. so number states. derive explicit formula for identify some properties \mapsto h_c(\beta ↦ Interestingly, latter necessarily monotone, meaning may be re-entrant.