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

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.