Parisi Functional

The Parisi functional is a variational functional encoding hierarchical overlap parameters of a mean-field spin glass. In the zero-temperature Gaussian Sherrington-Kirkpatrick convention with covariance , admissible parameters are nonnegative nondecreasing right-continuous functions on with . The value function solves

with terminal value . The functional is

Its minimum is the limiting expected maximum energy per spin of this model. Define the Stieltjes measure by . It is locally finite but need not be a probability measure or have finite total mass.