Negative correlations in combinatorics and statistical mechanics

1847年，基尔霍夫推导出了一个公式，表示电网的有效电导是其每个分支元件电导的函数。该公式是两个多项式的商，其中分子和分母编码有关网络的最小连通子图的信息。因此，电网络的物理特性可以被解释为对这些最小连接子图或生成树的定量陈述。例如，如果某个支路的电导增加，那么整个网络的有效电导不会减少。这就是所谓的“瑞利单调性”。它可以解释为，对于任意两个不同的分支，如果一个人随机选择一棵生成树，那么这些分支是“负相关的”——其中一个分支出现在树中会使另一个分支出现的可能性降低。这种现象的一种类比已经被证明适用于比电子网络更普遍的抽象组合几何。

In 1847, Kirchhoff derived a formula for the effective conductance of an electrical network as a function of the conductances of each of its branch elements. This formula is a quotient of two polynomials, in which the numerator and denominator encode information about the minimally connected subgraphs of the network. The physical properties of electrical networks can thus be interepreted as quantitative statements about these minimally connected subgraphs, or spanning trees. For example, if the conductance of some branch is increased then the effective conductance of the whole network does not decrease. This is known as ``Rayleigh monotonicity''. It translates into the statement that, for any two distinct branches, if one chooses a spanning tree at random then these branches are ``negatively correlated'' -- the presence of one of them in the tree makes the other one less likely. An analogue of this phenomenon has been shown to hold for abstract combinatorial geometries much more general than electrical networks.