Mechanism for the failure of the Edwards hypothesis in the Sherrington-Kirkpatrick spin glass

The dynamics of the Sherrington-Kirkpatrick 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(λ), is small at λ=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 modeling 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.