标题： 随机簇模型中$q=1$的相关性 显示英文标题

标题： Correlations in random cluster model at $q=1$

摘要： 设$\mu$是一个对有限基集的子集进行采样的测度，令$\mathcal{A}_e$表示元素$e$被采样的事件。 测度$\mu$如果对于任意一对元素$e, f$都有$\mu(\mathcal{A}_e \cap \mathcal{A}_f) - \mu(\mathcal{A}_e) \mu(\mathcal{A}_f) \leq 0$，则其为负相关。 如果不等式的方向相反，则该测度为正相关。 对于图上的随机簇模型，由于FKG不等式，已知边之间在$q \geq 1$时存在正相关性，而在$0 \leq q \leq 1$时负相关性仅是猜想。本文的主要结果是给出在$q=1$时有关差值的组合公式。之前，这样的公式仅在均匀生成树情况下已知，这是随机簇模型在$q=0$时的极限情况。 显示英文摘要

摘要： Let $\mu$ be a measure that samples a subset of a finite ground set, and let $\mathcal{A}_e$ be the event that element $e$ is sampled. The measure $\mu$ is negatively correlated if for any pair of elements $e, f$ one has $\mu(\mathcal{A}_e \cap \mathcal{A}_f) - \mu(\mathcal{A}_e) \mu(\mathcal{A}_f) \leq 0$. A measure is positively correlated if the direction of the inequality is reversed. For the random cluster model on graphs positive correlation between edges is known for $q \geq 1$ due to the FKG inequality, while the negative correlation is only conjectured for $0 \leq q \leq 1$. The main result of this paper is to give a combinatorial formula for the difference in question at $q=1$. Previously, such a formula was known in the uniform spanning tree case, which is a limit of the random cluster model at $q=0$.