Mean-field solution of the Potts glass near the transition temperature to the ordered phase

TL;DR

This paper analyzes the asymptotic mean-field solutions of the $p<4$ Potts glass near the transition temperature, revealing coexistence of degenerate ordered phases with different replica-symmetry breaking structures.

Contribution

It provides a detailed asymptotic expansion of solutions with various levels of replica-symmetry breaking for the Potts glass near the transition, identifying coexistence and stability properties.

Findings

Ordered phase is degenerate with multiple solutions.

Solutions with one hierarchy and continuous RSB coexist for p > 2.82.

Continuous RSB solution is marginally stable with highest free energy.

Abstract

We expand asymptotically mean-field solutions of the $p<4$ Potts glass with various levels of replica-symmetry breaking below the transition temperature to the glassy phase. We find that the ordered phase is degenerate and solutions with one hierarchy of spin replicas and with the full continuous replica-symmetry breaking coexist for $p> p^{*} \approx 2.82$. The latter emerges immediately with the instability of the replica-symmetric one. Apart from these two solutions there exists also a succession of unstable states converging to the solution with the continuous replica-symmetry breaking that is marginally stable and has the highest free energy.