A Percolation-theoretic Approach to Spin Glass Phase Transitions

The magnetically ordered, low temperature phase of Ising ferromagnets 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 tworeplica FK representation and also in the different Chayes—Machta—Redner two-replica graphical representation. Numerical studies of the ±

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.