Free energy of bipartite Sherrington-Kirkpatrick model

TL;DR

This paper extends the Parisi formula to the bipartite Sherrington-Kirkpatrick model, introducing a new overlap and analyzing the stability of replica symmetry, revealing a novel phase with partial symmetry breaking.

Contribution

It proves the free energy formula for the bipartite SK model, incorporating a new overlap and analyzing phase stability with partial replica symmetry breaking.

Findings

The free energy density is given by an extended Parisi formula.

A new phase with partial replica symmetry breaking is identified.

The stability of the replica symmetric solution is analyzed.

Abstract

In this paper we study the bipartite version of Sherrington-Kirkpatrick model. We prove that the free energy density is given by an analogue of the Parisi formula, that contains both the usual overlap and an additional new type of overlap. Following Panchenko, we prove the upper bound of the formula by Guerra's replica symmetry breaking interpolation, then the matching lower bound by Ghirlanda-Guerra identities and the Aizenman-Sims-Starr scheme. Based on this result we study the stability of the replica symmetric solution. A new phase exhibiting partial replica symmetry breaking is observed, where the broken phase is realized in the larger group only.