Pith Number

pith:NZ4OOK6H

pith:2026:NZ4OOK6HEWZMQXFFZQA4SGR6KE

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

Yan Ru Pei

The blue subgraph in the two-replica CMR representation of the Edwards-Anderson spin glass has at most two infinite connected components.

arxiv:2605.17338 v1 · 2026-05-17 · cond-mat.dis-nn · math-ph · math.MP

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Claims

C1strongest claim

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.

C2weakest assumption

The blue-bond process lacks insertion tolerance and positive association, so the proof must rely on the full joint measure together with a finite-box merge operation and the mass-transport bound on ends of translation-invariant subgraphs rather than standard Burton-Keane or random-cluster arguments (abstract, paragraph on main obstacle).

C3one line summary

Proves that the blue subgraph in the CMR representation of the short-range Edwards-Anderson model has at most two infinite clusters, which must lie in one grey cluster with opposite overlap parity if both are infinite.

References

12 extracted · 12 resolved · 0 Pith anchors

[1] Oliver Redner, Jon Machta, and Lincoln F. Chayes. Graphical representations and cluster algorithms for critical points with fields.Physical Review E, 58(3):2749–2752, 1998 1998

[2] J. Machta, C. M. Newman, and D. L. Stein. The percolation signature of the spin glass transition.Journal of Statistical Physics, 130:113–128, 2007 2007

[3] Cluster percolation in the two-dimensional Ising spin glass.Physical Review E, 107(5):054103, 2023 2023

[4] Cluster percolation in the three-dimensional±Jrandom- bond Ising model.Physical Review E, 113(2):024139, 2026 2026

[5] Graphical representations for Ising systems in external fields.Journal of Statistical Physics, 93:17–32, 1998 1998

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T00:03:52.964425Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6e78e72bc725b2c85ca5cc01c91a3e5132aa2eff1b9f94e669fe4f4eccea9ee9

Aliases

arxiv: 2605.17338 · arxiv_version: 2605.17338v1 · doi: 10.48550/arxiv.2605.17338 · pith_short_12: NZ4OOK6HEWZM · pith_short_16: NZ4OOK6HEWZMQXFF · pith_short_8: NZ4OOK6H

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/NZ4OOK6HEWZMQXFFZQA4SGR6KE \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 6e78e72bc725b2c85ca5cc01c91a3e5132aa2eff1b9f94e669fe4f4eccea9ee9

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "47bd3e21ac064883749f980709cda171fbd6a381c74c68875dde26f7090815b5",
    "cross_cats_sorted": [
      "math-ph",
      "math.MP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cond-mat.dis-nn",
    "submitted_at": "2026-05-17T09:15:39Z",
    "title_canon_sha256": "f778baab8c051b7ff29fb83fcb3d76e4f34e28b748f21b690294047683af7a1b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17338",
    "kind": "arxiv",
    "version": 1
  }
}