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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.17338 |
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| _version_ | 1866914575225454592 |
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| author | Pei, Yan Ru |
| author_facet | Pei, Yan Ru |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17338 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | At Most Two Infinite Blue Clusters in the CMR Representation of the Edwards-Anderson Spin Glass Pei, Yan Ru Disordered Systems and Neural Networks Mathematical Physics 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. |
| title | At Most Two Infinite Blue Clusters in the CMR Representation of the Edwards-Anderson Spin Glass |
| topic | Disordered Systems and Neural Networks Mathematical Physics |
| url | https://arxiv.org/abs/2605.17338 |