An interacting diffusion model and SK spin glass equation

Abstract

In this paper we consider a system of N interacting diffusion processes described by Itô stochastic differential equations. We first obtain a McKean-Vlasov limit for the empirical measure associated with the system in the limit as N → ∞. We then consider a special model (which has a “temperature” parameter β > 0) and show that the limiting process exhibits a phase transition phenomenon: for low temperatures (β > β c) it has a unique stable invariant measure while for high temperatures (β ≤ β c) the only invariant measure is the degenerate one. The former is a zero mean Gaussian measure such that its variance solves Sherrington-Kirkpatrick spin glass fixed point equation.