Metastability for Glauber Dynamics on the Complete Graph with Coupling Disorder

Communications in Mathematical Physics, vol. 392, pp. 307–345

Abstract

Abstract Consider the complete graph on n vertices. To each vertex assign an Ising spin that can take the values $$-1$$ - 1 or $$+1$$ + 1 . Each spin $$i \in [n]=\{1,2,\dots , n\}$$ i ∈ [ n ] = { 1 , 2 , ⋯ , n } interacts with a magnetic field $$h \in [0,\infty )$$ h ∈ [ 0 , ∞ ) , while each pair of spins $$i,j \in [n]$$ i , j ∈ [ n ] interact with each other at coupling strength $$n^{-1} J(i)J(j)$$ n - 1 J ( i ) J ( j ) , where $$J=(J(i))_{i \in [n]}$$ J = ( J ( i ) ) i ∈ [ 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 $$\beta \in (0,\infty )$$ β ∈ ( 0 , ∞ ) . We show that there are critical thresholds $$\beta _c$$ β c and $$h_c(\beta )$$ h c ( β ) such that, in the limit as $$n\rightarrow \infty $$ n → ∞ , the system exhibits metastable behaviour if and only if $$\beta \in (\beta _c, \infty )$$ β ∈ ( β c , ∞ ) and $$h \in [0,h_c(\beta ))$$ h ∈ [ 0 , h c ( β ) ) . Our main result is a sharp asymptotics, up to a multiplicative error $$1+o_n(1)$$ 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 $$\sqrt{n}$$

Authors 3

1. Anton Bovier

   University of Bonn

   Affiliation as printed

   Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

2. Frank den Hollander Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, P. O. Box 9512, 2300 RA, Leiden, The Netherlands

3. Saeda Marello corresponding

   University of Bonn

   Affiliation as printed

   Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany