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 $0<m=\mathbb{E}|J|<\infty$, the first blue-percolation inverse temperature ${\beta }_{\mathrm{b}}(d)$ satisfies $2dm{\beta }_{\mathrm{b}}(d)\to 1$ as $d\to \infty$, in periodic joint limits. For symmetric $\pm 1$ 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.

arXiv comment100 pages, 8 figures. v2: explicit unit-coupling dimensions 7-12 (v1: 22), finite spin-glass susceptibility and exact sector balance at those points, and imbalance implies ordinary overlap order. Companion code: https://github.com/PeaBrane/cmr-blue-percolation