The Percolation Signature of the Spin Glass Transition

TL;DR

This paper investigates the percolation signatures of spin glass transitions in Ising models, revealing that the transition is characterized by the emergence of two percolating clusters of unequal densities, using numerical and rigorous methods.

Contribution

It introduces a two-replica FK and graphical representation approach to identify the spin glass transition through percolation phenomena.

Findings

Spin glass transition corresponds to two percolating clusters of unequal densities.

Numerical studies support the percolation signature in the EA model.

Rigorous results confirm the percolation signature in the SK model.

Abstract

Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering -- both in short-range (EA) and infinite-range (SK) models -- within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the $\pm J$ EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of {\it two} percolating clusters of {\it unequal} densities.