An interacting diffusion model and SK spin glass equation
type: Item
instance of: scholarly article
title: An interacting diffusion model and SK spin glass equation
zbMATH Open document ID: 0818.60093
DOI: 10.1016/0304-4149(94)00025-5
author: Xinhong Ding
published in: Stochastic Processes and their Applications
publication date: 1995-08-07T00:00:00Z
review text: The author studies a system of \(N\) interacting diffusions related to a spin glass model of \textit{Sherrington} and \textit{Kirkpatrick} (SK) [Phys. Rev. Lett. 35, 1792-1795 (1975)]. He proves, that the empirical measure of the system converges, as \(N\to\infty\), to a deterministic measure. It is the distribution of a process determined by a single equation and satisfies a McKean-Vlasov equation in the weak sense. He then considers a special case, in which the drift term depends on a parameter \(\beta > 0\), which plays a similar role as the inverse temperature in equilibrium statistical mechanics. This model is related to a random neural network model of \textit{Sompolinsky} and \textit{Crisanti} [ibid. 61, 259-262 (1988)]. It is shown, that the limit process exhibits a phase transition: there exists a critical \(\beta_ c > 0\), such that for high temperatures \(\beta \leq \beta_ c\) there are no nontrivial invariant measures, whereas for low temperatures \(\beta > \beta_ c\) there exists a unique nontrivial invariant measure. This is Gaussian and its covariance solves the SK-spin glass fixed point equation.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 724744
zbMATH Keywords: interacting diffusions
zbMATH Keywords: spin glass model
zbMATH Keywords: empirical measure
zbMATH Keywords: McKean- Vlasov equation
zbMATH Keywords: random neural network model
zbMATH Keywords: fixed point equation
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)00025-5
OpenAlex ID: W2059133277
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description: scientific article; zbMATH DE number 724744