发表机构
Tianjin University（天津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明高正则图序列在随机团模型参数低于一时，边对具有严格负相关性，并建立了与一般图猜想的等价性，通过聚合物表示和簇展开方法实现。
AI 中文摘要
具有团参数 \\(0<Q<1\\) 的随机团模型被猜想表现出负依赖性，但即使在一般有限图上，不同边之间的成对负相关性也仍然是开放的。我们首先表明，将问题限制在度数发散的规则图上并不会实质性地削弱成对猜想：对所有此类图序列的有效性，即使仅限于距离为 \\(o(d/\log d)\\) 的边对，也等价于在任意有限图上的有效性。然后，我们证明了一类满足双尺度边等周条件的广义高正则图序列具有严格的成对负相关性。更精确地说，对于每个固定的 \\(p\in(0,1)\\)，序列中所有足够大的图，其任意两个不同边指示变量在距离为 \\(o(d/\log d)\\) 时具有严格负协方差，且该结果在 \\(Q\in[0,1)\\) 和两条边的选择上一致成立。特别地，该结果包括 \\(Q=0\\) 端点，对应于以连通为条件的伯努利键渗流。等周假设由高正则展开图族满足，并且对于度数发散的均匀随机正则图，以趋于一的概率满足；其双尺度形式也允许乘积几何，如超立方体和固定边长的高维离散环面。证明基于聚合物表示和簇展开，以及对两边缘相关性主要贡献的几何识别。
英文摘要
The random-cluster model with cluster parameter \(0<q<1\) is conjectured to exhibit negative dependence, but even pairwise negative correlation between distinct edges remains open on general finite graphs. We first establish a reduction showing that the pairwise conjecture is equivalent to its restriction to regular graph sequences of diverging degrees $(d_n)$, even for edge pairs at distance \(o(d_n/\log d_n)\). We then prove strict pairwise negative correlation for a broad class of high-degree regular graph sequences satisfying a two-scale edge-isoperimetric condition. More precisely, for every fixed \(p\in(0,1)\), all sufficiently large graphs in the sequence have strictly negative covariance between any two distinct edge indicators at distance \(o(d_n/\log d_n)\), uniformly in \(q\in[0,1)\) and in the choice of the two edges. In particular, the result includes the \(q=0\) endpoint, corresponding to Bernoulli bond percolation conditioned to be connected. The isoperimetric hypothesis is satisfied by high-degree expander families and, with probability tending to one, by uniformly random regular graphs of diverging degree; its two-scale form also allows product geometries such as hypercubes and fixed-side high-dimensional discrete tori. The proof is based on a polymer representation and cluster expansion, together with a geometric identification of the leading contribution to the two-edge correlation.
CommentsProposition 6.5 has been modified, and relevant references have been added
