AI 中文总结
本文通过Chayes-Machta-Redner表示中的蓝色渗流，证明了高维Edwards-Anderson自旋玻璃的渗流起始渐近行为，并给出维度22的计算机辅助证明，为理解自旋玻璃相变提供了新途径。
AI 中文摘要
我们证明了在高有限维度下，零场Edwards-Anderson自旋玻璃的Chayes-Machta-Redner表示中的蓝色渗流。对于固定的对称独立同分布耦合，满足$0<m=\mathbb{E}|J|<\infty$，在周期联合极限下，首次渗流起始渐近于$(2dm)^{-1}$。对于对称$\pm1$耦合，我们给出了一个计算机辅助的维度22证书。对于指数矩无序，我们还建立了一个渗流区域，其中无限蓝色扇区密度完全相等，且自旋玻璃磁化率有限。在这些周期极限中，持续的不平衡密度将意味着不同的自旋翻转相关的Gibbs态，以及对于合适的有限局域复制耦合出现压力尖点，而无需有限蓝色抵消。证明不平衡仍然是一个开放问题。
英文摘要
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 $β_{\mathrm{b}}(d)$ satisfies $2dm\,β_{\mathrm{b}}(d)\to1$ as $d\to\infty$, in periodic joint limits. For symmetric $\pm1$ 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.
Comments100 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
