Stochastic equations and dynamics beyond mean-field theory

TL;DR

This paper derives and numerically solves stochastic equations describing critical fluctuations in spin-glass models beyond mean-field theory, showing excellent agreement with simulations.

Contribution

It analytically computes parameters of dynamical stochastic equations for finite-dimensional spin glasses using the cavity method and validates predictions against numerical data.

Findings

Excellent match between stochastic equation predictions and simulations

Quantitative parameters derived analytically for 3-spin Bethe lattice

Critical fluctuations smoothed out in finite systems and dimensions

Abstract

The dynamical transition occurring in spin-glass models with one step of Replica-Symmetry-Breaking is a mean-field artifact that disappears in finite systems and/or in finite dimensions. The critical fluctuations that smooth the transition are described in the $\beta$ regime by dynamical stochastic equations. The quantitative parameters of the dynamical stochastic equations have been computed analytically on the 3-spin Bethe lattice Spin-Glass by means of the (static) cavity method and the equations have been solved numerically. The resulting parameter-free dynamical predictions are shown here to be in excellent agreement with numerical simulation data for the correlation and its fluctuations.