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Color symmetry in the Potts spin glass at high temperature

Heejune Kim

Abstract

We show that color symmetry is preserved at high temperatures in the Potts spin glass model with $κ\ge 3$ 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 $κ= 2$, we exploit the model's gauge symmetry to show that unbalanced configurations occur with exponentially small probability at all temperatures $β\in [0, \infty]$.

Color symmetry in the Potts spin glass at high temperature

Abstract

We show that color symmetry is preserved at high temperatures in the Potts spin glass model with 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 , we exploit the model's gauge symmetry to show that unbalanced configurations occur with exponentially small probability at all temperatures .
Paper Structure (9 sections, 12 theorems, 88 equations)

This paper contains 9 sections, 12 theorems, 88 equations.

Table of Contents

  1. Introduction and main results
  2. Open problems.
  3. Proof sketch
  4. Related work.
  5. Proof of Theorem \ref{['thm: high temp']}
  6. Proof of Proposition \ref{['prop: kappa=2']}
  7. Necessity of Hamiltonian centering
  8. Deferred proofs
  9. Color symmetry breaking at zero temperature

Key Result

Theorem 1.3

For any $\kappa\ge 3$ and $\beta \in [0, \beta_\kappa)$, we have In particular, color symmetry is preserved in this high temperature regime.

Theorems & Definitions (28)

  • Definition 1.1: Color symmetry
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Proposition 2.1
  • proof : Proof of Theorem \ref{['thm: high temp']}
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 18 more