An interacting diffusion model and SK spin glass equation (Q1344949)

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.