On the Stability of the Quenched State in Mean Field Spin Glass Models

TL;DR

This paper investigates the stability of the quenched state in mean field spin glass models, linking temperature continuity assumptions to stability conditions and overlap distribution constraints, with implications for the Parisi Ansatz.

Contribution

It establishes a connection between temperature continuity, stability conditions, and overlap polynomial vanishing in mean field spin glasses, extending the Parisi Ansatz framework.

Findings

Stability conditions are equivalent to marginal additivity of quenched free energy.

Continuity assumptions impose constraints on overlap distributions.

Overlap polynomials can be computed using a zero-replica limit.

Abstract

While the Gibbs states of spin glass models have been noted to have an erratic dependence on temperature, one may expect the mean over the disorder to produce a continuously varying ``quenched state''. The assumption of such continuity in temperature implies that in the infinite volume limit the state is stable under a class of deformations of the Gibbs measure. The condition is satisfied by the Parisi Ansatz, along with an even broader stationarity property. The stability conditions have equivalent expressions as marginal additivity of the quenched free energy. Implications of the continuity assumption include constraints on the overlap distribution, which are expressed as the vanishing of the expectation value for an infinite collection of multi-overlap polynomials. The polynomials can be computed with the aid of a "real"-replica calculation in which the number of replicas is taken to zero.