Negative correlations in combinatorics and statistical mechanics

1847年，基尔霍夫推导出了一个网络有效电导的公式，它是每个支路元件的电导的函数。这个公式是两个多项式的商，其中分子和分母编码关于网络的最小连通子图的信息。因此，电网络的物理属性可以被解释为关于这些最小连通子图或生成树的定量陈述。例如，如果某个支路的电导增加，则整个网络的有效电导不会降低。这就是众所周知的“瑞利单调性”。它转化为这样一种说法，对于任何两个不同的分支，如果随机选择一个生成树，那么这些分支是“负相关的”--其中一个出现在树中会降低另一个的可能性。这一现象的类比已被证明适用于比电网络更一般的抽象组合几何。

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.