Percolation in the Sherrington-Kirkpatrick Spin Glass

TL;DR

This paper investigates percolation phenomena in the Sherrington-Kirkpatrick spin glass model, revealing how large percolating clusters relate to ultrametricity and the spin overlap distribution, with detailed proofs and theoretical insights.

Contribution

It provides extended proofs and detailed analysis of percolation in SK spin glasses, linking cluster densities to ultrametricity and spin overlaps, advancing theoretical understanding.

Findings

Ordered phase has a unique maximal percolating cluster.

Percolating cluster densities reflect ultrametricity.

Connection established between cluster densities and spin overlap distribution.

Abstract

We present extended versions and give detailed proofs of results concerning percolation (using various sets of two-replica bond occupation variables) in Sherrington-Kirkpatrick spin glasses (with zero external field) that were first given in an earlier paper by the same authors. We also explain how ultrametricity is manifested by the densities of large percolating clusters. Our main theorems concern the connection between these densities and the usual spin overlap distribution. Their corollaries are that the ordered spin glass phase is characterized by a unique percolating cluster of maximal density (normally coexisting with a second cluster of nonzero but lower density). The proofs involve comparison inequalities between SK multireplica bond occupation variables and the independent variables of standard Erdos-Renyi random graphs.