Potts Glass on Random Graphs

TL;DR

This paper analyzes the q-state Potts model with anti-ferromagnetic interactions on large random graphs, revealing glassy behavior and phase transitions similar to structural glasses using the cavity method.

Contribution

It introduces a cavity method analysis of the Potts glass on random graphs, detailing phase diagrams and stability of replica symmetry breaking solutions.

Findings

Model exhibits mean field spin glass behavior due to frustration.

Identifies dynamic and static glass transition points.

Shows stability of one-step RSB in the colorable region.

Abstract

We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves as a mean field spin glass. We use the cavity method to compute the temperature-coordination phase diagram and to determine the location of the dynamic and static glass transitions, and of the Gardner instability. We show that for q>=4 the model possesses a phenomenology similar to the one observed in structural glasses. We also illustrate the links between the positive and the zero-temperature cavity approaches, and discuss the consequences for the coloring of random graphs. In particular we argue that in the colorable region the one-step replica symmetry breaking solution is stable towards more steps of replica symmetry breaking.