格点二聚体模型维里系数的正性及图的匹配数的上界
Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs
摘要：
利用无穷正则格点上单体-二聚体模型中压力的维里膨胀系数与熵膨胀系数之间的关系，我们证明了在任意维的超立方格点上，维里系数在20阶以上都是正的.我们已经观察到，所有维里系数迄今已知的这个系统是积极的也具有不同结构的无限规则格。因此我们推测维里展开系数$m_k $总是正的。这些考虑可以推广到有限图的相关界的研究，推广到无限正则格，即有限格和正则双连通图。的有效性的界限$Δ^k {\rm ln}(i! N(i)) \le 0$为$k \ge 2$，其中$N(i)$是图上的$i$二聚体的配置数和$Δ$是向前的差异运营商，示出对应于维里系数的积极性。我们在许多有限格图上的测试表明，在大格上这些界限是满足的，支持维里系数的正性的猜想。一个详尽的调查，一些类的定期双连通图与一个不太大的数量$v$的顶点显示只有少数违反这些界限。我们推测，违规的频率为$v \to \infty$消失。我们发现严格的上界$N(i)$有效的任意图和正则图。Heilman-Lieb不等式与上述不等式的相似性表明，我们可以研究维里系数的更严格的不等式$m_k \ge \frac{1}{2k}$，它适用于无限正则格点模型的所有已知系数.
英文摘要：
Using a relation between the virial expansion coefficients of the pressure and the entropy expansion coefficients in the case of the monomer-dimer model on infinite regular lattices, we have shown that, on hypercubic lattices of any dimension, the virial coefficients are positive through the 20th order. We have observed that all virial coefficients so far known for this system are positive also on infinite regular lattices with different structure. We are thus led to conjecture that the virial expansion coefficients $m_k $ are always positive. These considerations can be extended to the study of related bounds on finite graphs generalizing the infinite regular lattices, namely the finite grids and the regular biconnected graphs. The validity of the bounds $Δ^k {\rm ln}(i! N(i)) \le 0$ for $k \ge 2$, where $N(i)$ is the number of configurations of $i$ dimers on the graph and $Δ$ is the forward difference operator, is shown to correspond to the positivity of the virial coefficients. Our tests on many finite lattice graphs indicate that on large lattices these bounds are satisfied, giving support to the conjecture on the positivity of the virial coefficients. An exhaustive survey of some classes of regular biconnected graphs with a not too large number $v$ of vertices shows only few violations of these bounds. We conjecture that the frequency of the violations vanishes as $v \to \infty$. We find rigorous upper bounds on $N(i)$ valid for arbitrary graphs and for regular graphs. The similarity between the Heilman-Lieb inequality and the one conjectured above suggests that one study the stricter inequality $m_k \ge \frac{1}{2k}$ for the virial coefficients, which is valid for all the known coefficients of the infinite regular lattice models.
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