At Most Two Infinite Blue Clusters in the CMR Representation of the Edwards-Anderson Spin Glass

TL;DR

This paper proves a structural constraint in the CMR representation of the Edwards-Anderson spin glass, showing at most two infinite blue clusters can exist, supporting the two-cluster spin-glass phase picture.

Contribution

It establishes a rigorous upper bound on the number of infinite blue clusters in the CMR representation of short-range spin glasses.

Findings

Blue subgraph contains at most two infinite connected components.

If two infinite blue clusters exist, they are in opposite overlap-parity classes.

Finite energy and percolation transition are established for the grey subgraph.

Abstract

The two-replica Chayes-Machta-Redner (CMR) representation is one of the main proposed geometric signatures of spin-glass order in the short-range Edwards-Anderson model. Mean-field arguments and recent numerics suggest that the low-temperature phase should exhibit two macroscopic blue clusters carrying opposite overlap signs. We prove a rigorous structural constraint in this direction. For any translation-invariant joint Gibbs measure on disorder, two spin replicas, and CMR bond variables on Z^d, the blue subgraph contains at most two infinite connected components; if two exist, then they lie in a common infinite grey cluster and belong to opposite overlap-parity classes. The main obstacle is that the blue-bond process is neither insertion-tolerant nor positively associated, so the usual Burton-Keane and random-cluster arguments do not apply. We circumvent this by working in the full joint measure and using a finite-box merge operation together with the mass-transport bound on ends of translation-invariant subgraphs. As auxiliary input, we establish finite energy and a percolation transition for the grey subgraph via Bayesian resampling of couplings and a parity-based Peierls estimate. These results do not prove the existence of infinite blue clusters or a spin-glass phase transition, but they give a rigorous upper bound compatible with the two-cluster picture for short-range spin glasses.