Towards the spin-glass transition in finite dimensions via blue percolation
Abstract
We prove blue percolation in the Chayes-Machta-Redner representation of the zero-field Edwards-Anderson spin glass in finite dimensions. For fixed symmetric iid couplings with 0<m=𝔼|J|<∞, the first blue-percolation inverse temperature βb(d) satisfies 2dm βb(d)→1 as d→∞, in periodic joint limits. For symmetric ±1 couplings, both overlap signs percolate at explicit temperatures in every dimension from 7 to 12. That proof reveals the overlap field, shows that it dominates a weakly coupled Ising field, and bounds a weighted second moment of open paths; in dimensions 7 to 9 the paths make short lateral excursions, and an interaction-matrix criterion controls the second moment. These proofs are computer-assisted: finitely many inequalities between rigorously bounded quantities are decided in exact rational arithmetic. For exponential-moment disorder, we also establish a percolating regime with exactly equal infinite-blue-sector densities and finite spin-glass susceptibility. In these periodic limits, persistent density imbalance would imply distinct spin-flip-related Gibbs states and ordinary spin-glass order: positive spatial-overlap variance, infinite spin-glass susceptibility and a cusp of the replica-coupling pressure, without requiring finite-blue cancellation. Proving imbalance remains open.
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BibTeX
@misc{pei2026towards,
title = {{Towards the spin-glass transition in finite dimensions via blue percolation}},
author = {Yan Ru Pei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22301},
eprint = {2609.22301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22301},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Towards the spin-glass transition in finite dimensions via blue percolation
AU - Yan Ru Pei
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.22301
UR - https://arxiv.org/abs/2609.22301
AB - We prove blue percolation in the Chayes-Machta-Redner representation of the zero-field Edwards-Anderson spin glass in finite dimensions. For fixed symmetric iid couplings with 0<m=𝔼|J|<∞, the first blue-percolation inverse temperature β_b(d) satisfies 2dm β_b(d)→1 as d→∞, in periodic joint limits. For symmetric ±1 couplings, both overlap signs percolate at explicit temperatures in every dimension from 7 to 12. That proof reveals the overlap field, shows that it dominates a weakly coupled Ising field, and bounds a weighted second moment of open paths; in dimensions 7 to 9 the paths make short lateral excursions, and an interaction-matrix criterion controls the second moment. These proofs are computer-assisted: finitely many inequalities between rigorously bounded quantities are decided in exact rational arithmetic. For exponential-moment disorder, we also establish a percolating regime with exactly equal infinite-blue-sector densities and finite spin-glass susceptibility. In these periodic limits, persistent density imbalance would imply distinct spin-flip-related Gibbs states and ordinary spin-glass order: positive spatial-overlap variance, infinite spin-glass susceptibility and a cusp of the replica-coupling pressure, without requiring finite-blue cancellation. Proving imbalance remains open.
ER -