Negative correlations in combinatorics and statistical mechanics

组合数学和统计力学中的负相关

• 批准号: 105392-2007
• 负责人: Wagner, David
• 金额: $ 1.24万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2010
• 资助国家: 加拿大
• 起止时间: 2010-01-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=451257
• 关键词: Negative; correlations; combinatorics; statistical; mechanics

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.

在1847年，基尔霍夫推导出了一个公式，表示一个电网络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关于网络的最小连通子图的信息。 因此，电网络的物理性质可以理解为关于这些最小连通子图或生成树的定量陈述。 例如，如果某个分支的电导增加，则整个网络的有效电导不会减少。 这被称为“Rayleigh单调性”。 它转化为这样的陈述，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”-其中一个在树中的存在使得另一个不太可能。 这种现象的一个类似物已被证明适用于比电网络更一般的抽象组合几何。