On the absence of phase transition in the monomer-dimer model
On the absence of phase transition in the monomer-dimer model
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关于单体-二聚体模型中不存在相变的问题
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发表时间：
1998
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通讯作者：
J. Berg
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作者：
J. Berg
假设我们用不重叠的单体(单体集)和二聚体(对应于一条边的顶点对)覆盖图$G$的顶点集。每种方法都被称为单体-二聚体构型。如果$G$是有限的，且$lambda<；0$，我们将$G$(参数为$lambda$)的单体-二聚体分布定义为为每个单体-二聚体构型分配与$lambda^{|Mbox{dimer}|}$成比例的概率分布，其中$|mbox{dimer}|$是该构型中的二聚体数目。如果图是无限的，则单体-二聚体分布可以通过采用弱极限以标准方式构造。我们特别感兴趣的是$d$维立方晶格(子图)上的单体-二聚体模型。海尔曼和利布(1972)用某些热力学函数的光滑性证明了不存在相变。他们通过研究配分函数零点在复平面中的位置来做到这一点。我们提出了一种不同的方法，并通过概率论证表明，当到边界的距离达到$infty$时，边界效应变得可以忽略不计。这在某种意义上没有相变，但通常不同于上述意义上的相变。然而，边界效应的衰减似乎以如此强烈的方式发生，以至于根据Dobrushin和Shlosman(1987)和Dobrushin和Warstat(1990)对一般Gibbs系统的结果，热力学函数的光滑性随之而来。更准确地说，我们证明了，在他们的术语中，该模型是{em完全解析的}。
Suppose we cover the set of vertices of a graph $G$ by non-overlapping monomers (singleton sets) and dimers (pairs of vertices corresponding to an edge). Each way to do this is called a monomer-dimer configuration. If $G$ is finite and $lambda < 0$, we define the monomer-dimer distribution for $G$ (with parameter $lambda$) as the probability distribution which assigns to each monomer-dimer configuration a probability proportional to $lambda^{|mbox{dimers}|}$, where $|mbox{dimers}|$ is the number of dimers in that configuration. If the graph is infinite, monomer-dimer distributions can be constructed in the standard way, by taking weak limits. We are particularly interested in the monomer-dimer model on (subgraphs of) the $d$-dimensional cubic lattice. Heilmann and Lieb (1972) prove absence of phase transition, in terms of smoothness properties of certain thermodynamic functions. They do this by studying the location in the complex plane of the zeros of the partition function. We present a different approach and show, by probabilistic arguments, that boundary effects become negligible as the distance to the boundary goes to $infty$. This gives absence of phase transition in a related, but generally not equivalent sense as above. However, the decay of boundary effects appears to occur in such a strong way that, by results on general Gibbs systems of Dobrushin and Shlosman (1987) and Dobrushin and Warstat (1990), smoothness properties of thermodynamic functions follow. More precisely we show that, in their terminology, the model is {em completely analytic}.