A Percolation-Theoretic Approach to Spin Glass Phase Transitions

TL;DR

This paper reviews recent work on identifying spin glass phase transitions through percolation signatures in various models, highlighting the role of percolating clusters in detecting spin-glass order.

Contribution

It introduces a percolation-theoretic framework for detecting spin glass phases using two-replica FK representations, extending previous methods for ferromagnets.

Findings

Percolating clusters of maximal density indicate spin-glass order.

Numerical and rigorous results support the percolation signature in EA and SK models.

Presence of two percolating clusters of different densities characterizes the spin-glass phase.

Abstract

The magnetically ordered, low temperature phase of Ising ferro- magnets is manifested within the associated Fortuin-Kasteleyn (FK) random cluster representation by the occurrence of a single positive density percolating cluster. In this paper, we review our recent work on the percolation signature for Ising spin glass ordering -- both in the short-range Edwards-Anderson (EA) and infinite-range Sherrington-Kirkpatrick (SK) models -- within a two-replica FK representation and also in the different Chayes-Machta-Redner two-replica graphical representation. Numerical studies of the $\pm J$ EA model in dimension three and rigorous results for the SK model are consistent in supporting the conclusion that the signature of spin-glass order in these models is the existence of a single percolating cluster of maximal density normally coexisting with a second percolating cluster of lower density.