Spin-Glass Stochastic Stability: a Rigorous Proof

TL;DR

This paper provides a rigorous proof of stochastic stability in spin-glass models, confirming its validity in the thermodynamic limit and relating it to previous results by Ghirlanda and Guerra.

Contribution

It offers a rigorous proof of stochastic stability for the Sherrington-Kirkpatrick and Edwards-Anderson models, previously assumed but not proved.

Findings

Stochastic stability holds in beta-average for both models.

The volume rate of convergence is V^{-1}.

Stochastic stability identities match those by Ghirlanda and Guerra.

Abstract

We prove the property of stochastic stability previously introduced as a consequence of the (unproved) continuity hypothesis in the temperature of the spin-glass quenched state. We show that stochastic stability holds in beta-average for both the Sherrington-Kirkpatrick model in terms of the square of the overlap function and for the Edwards-Anderson model in terms of the bond overlap. We show that the volume rate at which the property is reached in the thermodynamic limit is V^{-1}. As a byproduct we show that the stochastic stability identities coincide with those obtained with a different method by Ghirlanda and Guerra when applyed to the thermal fluctuations only.