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

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 subsequential local weak limit of the standard periodic-torus joint laws on disorder, two spin replicas, and CMR bond variables, 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 labelled blue geometry does not permit unrestricted insertions across overlap classes, and no positive-association input is available, so the usual Burton-Keane and random-cluster arguments do not apply directly.

We isolate an abstract multicolour Burton-Keane proposition based on finite-box label-class coalescence and verify its hypothesis for CMR blue bonds by resampling the full joint measure.

As auxiliary input, we establish finite energy and a percolation transition for the grey subgraph via local resampling of the disorder 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.