Free energy in spin glass models with conventional order

TL;DR

This paper rigorously verifies a min-max formula for free energy in vector spin glass models with conventional order parameters, extending the Parisi formula to include additional deterministic interactions.

Contribution

It provides a rigorous proof of the min-max formula for free energy in vector spin glasses with conventional order, generalizing previous Parisi formula results.

Findings

Verification of the min-max formula for free energy

Extension of Parisi formula to models with additional order parameters

Applicability to vector spin glass models with deterministic interactions

Abstract

Recently, [DOI:10.1007/s10955-023-03135-1] considered spin glass models with additional conventional order parameters characterizing single-replica properties. These parameters are distinct from the standard order parameter, the overlap, used to measure correlations between replicas. A ``min-max'' formula for the free energy was prescribed in [DOI:10.1007/s10955-023-03135-1]. We rigorously verify this prescription in the setting of vector spin glass models featuring additional deterministic spin interactions. Notably, our results can be viewed as a generalization of the Parisi formula for vector spin glass models in [DOI:10.1214/17-AOP1194], where the order parameter for self-overlap is already present.