AN INTERACTING DIFFUSION-MODEL AND SK SPIN-GLASS EQUATION

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:54

Article

关键词: INTERACTING DIFFUSIONS;    MCKEAN-VLASOV LIMIT;    NEURAL NETWORKS;    SPIN GLASSES;   

DOI  :  10.1016/0304-4149(94)00025-5

来源: Elsevier

【 摘 要 】

In this paper we consider a system of N interacting diffusion processes described by Ito stochastic differential equations. We first obtain a McKean-Vlasov limit for the empirical measure associated with the system in the limit as N --> infinity. We then consider a special model (which has a ''temperature'' parameter beta > 0) and show that the limiting process exhibits a phase transition phenomenon: for low temperatures (beta > beta(c)) it has a unique stable invariant measure while for high temperatures (beta less than or equal to beta(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.

【 授权许可】

Free