发表机构
Umeå University（于默奥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明树状气体和 $q<1$ 随机团簇模型在特定条件下满足负边相关性，并探索了连通事件的正相关性，为统计物理中的相关性研究提供了新结果。
AI 中文摘要
我们研究了统计物理学中两个著名模型——树状气体和 $q<1$ 随机团簇模型——的负边相关性。我们证明了在边 $e$ 和 $f$ 被移除后，它们的瑞利差是一个交叉贡献减去两个端点连通事件的协方差。对于任意有限图上的树状气体，我们分析了这一差的两个首项系数。第一个是经典的传递电流平方。如果它为零，第二个具有电学平方和公式。这证明了当瑞利差的两个首项系数之一非零时，在足够大的逸度下负边相关性成立，并通过等势分解刻画了同时相等的情形。对于完全图 $K_n$ 上的随机团簇模型，我们证明了当所有边权至少为 $2$ 时，在整个 $q<1$ 范围内负边相关性成立。在均匀情形下，我们证明了更强的结论：当所有权相等且至少为 $1$ 时，负相关性成立。特别地，$K_n$ 的连通生成子图中的不同边，以固定逸度加权，对每个 $n$ 都是负相关的。我们还探索了一个似乎是普遍现象的问题：连通事件的正相关性。对于树状气体，我们证明了在高逸度下连通事件的正相关性。在格点上，推测的不等式将使两点函数具有超乘性，并产生凸的逆相关长度。精确计算支持图和拟阵猜想，包括 Seymour--Welsh 拟阵 $\mathcal S_8$，其中普通的边负相关性不成立。本文的结果由作者推导，未使用大型语言模型。作者确实受益于使用 GPT-5 Pro 生成代码来检验假设以及简化论证。
英文摘要
We study negative edge correlation for two well-known models in statistical physics, the arboreal gas and the $q < 1$ random-cluster model. We show that after the edges $e$ and $f$ are removed, their Rayleigh difference is a crossing contribution minus the covariance of two endpoint-connectivity events. For the arboreal gas on an arbitrary finite graph, we analyze the two leading coefficients of this difference. The first is the classical transfer-current square. If it vanishes, the second has an electrical sum-of-squares formula. This proves negative edge correlation at sufficiently large fugacity whenever one of the two leading coefficients of the Rayleigh difference is nonzero, and characterizes simultaneous equality by an equipotential decomposition. For the random-cluster model on the complete graph $K_n$, we prove negative edge correlation throughout $q<1$ when all edge weights are at least $2$. In the uniform case, we prove something stronger:negative correlation holds when all the weights are equal and at least $1$. In particular, distinct edges in a connected spanning subgraph of $K_n$, weighted by a fixed fugacity, are negatively correlated for every $n$. We also explore what seems to be a general phenomenon: positive correlation of connectivity events. For the arboreal gas, we prove positive correlation for connectivity events at high fugacities. On lattices, the conjectured inequality would make the two-point function supermultiplicative and produce a convex inverse correlation length. Exact computations support the graph and matroid conjectures, including the Seymour--Welsh matroid $\mathcal S_8$, where ordinary edge-negative correlation fails. The results in this paper were derived by the authors without the use of Large Language models. The authors did benefit from using GPT-5 Pro for generating code to test out hypotheses as well as for simplifying the arguments.
Comments45 pages, comments are welcome
