Metastability of Glauber dynamics with inhomogeneous coupling disorder

TL;DR

This paper studies the metastability behavior of a broad class of mean-field-like spin systems with random couplings, showing that their metastable properties can be approximated by the averaged system under certain conditions, extending known results for Erdős–Rényi graphs.

Contribution

It establishes conditions under which the metastability of complex inhomogeneous spin systems can be approximated by their averaged counterparts, extending previous results to more general models.

Findings

Metastability of complex systems is approximated by averaged systems with high probability.

Asymptotic tail behavior of metastable hitting times is characterized.

Results extend known metastability analysis from Erdős–Rényi to more general inhomogeneous models.

Abstract

We introduce a general class of mean-field-like spin systems with random couplings that comprises both the Ising model on inhomogeneous dense random graphs and the randomly diluted Hopfield model. We are interested in quantitative estimates of metastability in large volumes at fixed temperatures when these systems evolve according to a Glauber dynamics, i.e.\ where spins flip with Metropolis transition probabilities at inverse temperature $\beta$. We identify conditions ensuring that with high probability the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. This work provides an extension of the results known for the Ising model on the Erd\H{o}s--R{\'e}nyi random graph. The proofs use the potential-theoretic approach to metastability in combination with concentration inequalities.