M.Aizenman, P.Contucci
On the Stability of the Quenched State in Mean Field
Spin Glass Models
(39K, LATeX)
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.