Mechanism for the failure of the Edwards hypothesis in the SK spin glass

TL;DR

This paper investigates why the Edwards hypothesis fails in the SK spin glass model at zero temperature, showing that the metastable states reached by dynamics are atypical and modeling the energy changes as a Markov process to explain this failure.

Contribution

It demonstrates the qualitative failure of the Edwards hypothesis in the SK model and models the energy dynamics as a Markov process to explain this phenomenon.

Findings

Metastable states have low probability density at zero site energy.

Most metastable states differ significantly from typical states.

Markov process modeling reveals physical basis for the failure.

Abstract

The dynamics of the SK model at T=0 starting from random spin configurations is considered. The metastable states reached by such dynamics are atypical of such states as a whole, in that the probability density of site energies, $p(\lambda)$, is small at $\lambda=0$. Since virtually all metastable states have a much larger $p(0)$, this behavior demonstrates a qualitative failure of the Edwards hypothesis. We look for its origins by modelling the changes in the site energies during the dynamics as a Markov process. We show how the small $p(0)$ arises from features of the Markov process that have a clear physical basis in the spin-glass, and hence explain the failure of the Edwards hypothesis.