Color symmetry in the Potts spin glass at high temperature

TL;DR

This paper proves that color symmetry remains intact at high temperatures in the Potts spin glass model for three or more colors, using probabilistic methods and existing results from the non-disordered Potts model.

Contribution

It introduces a novel proof employing the second moment method and leverages gauge symmetry for the two-color case, advancing understanding of symmetry preservation in spin glasses.

Findings

Color symmetry preserved at high temperatures for κ ≥ 3

Unbalanced configurations are exponentially rare for κ=2

Utilizes second moment method and gauge symmetry techniques

Abstract

We show that color symmetry is preserved at high temperatures in the Potts spin glass model with $\kappa \ge 3$ colors. Our proof employs the second moment method applied to the balanced model with a suitable centering of the Hamiltonian, while incorporating results from the non-disordered Potts model Ellis--Wang (1990), https://doi.org/10.1016/0304-4149(90)90122-9. For $\kappa = 2$, we exploit the model's gauge symmetry to show that unbalanced configurations occur with exponentially small probability at all temperatures $\beta \in [0, \infty]$.