On Parisi measures of Potts spin glasses with correction

Abstract

In the Potts spin glass model, inspired by the symmetry argument in [4] for the constrained free energy, we study the free energy with self-overlap correction. Similarly, we simplify the Parisi-type formula, originally an infimum over matrix-valued paths, into an optimization over real-valued paths, which can be interpreted as quantile functions of probability measures on the unit interval. Minimizers are usually called Parisi measures. Using results in [9] generalizing the Auffinger–Chen convexity argument, we deduce the uniqueness of the Parisi measure.

The approach in [4] is to perturb the covariance function of the Hamiltonian into one associated with the generic model, prove the results there, and then reduce the perturbation. Here, we choose to perturb the model by adding an external field parametrized by Ruelle probability cascades. This is used in the Hamilton–Jacobi equation approach and we directly apply results in [12].