论文标题
平面图的笛卡尔产品的单极二聚体模型
The monopole-dimer model on Cartesian products of plane graphs
论文作者
论文摘要
单极二聚体模型是具有决定性结构的单体二聚体模型的签名变体。我们将平面图的单极二聚体模型（Math。Phys。Anal。Geom。，2015）扩展到其笛卡尔产品，并表明该模型的分区函数可以表示为通用签名的邻接矩阵的决定因素。然后，我们证明，只要方向是pfaffian，分区函数与平面图的方向无关。当这些平面图是双分部分时，我们表明分区函数的计算变得特别简单。然后，我们为三维网格图的分区函数提供明确的产品公式，即La Kasteleyn和Tembyley--Fischer，当所有网格长度均匀时，它是多项式的第四幂。最后，我们将此产品公式推广到$ d $尺寸，再次获得了明确的产品公式。最后，我们讨论了自由能和单极密度的渐近公式。
The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. We extend the monopole-dimer model for planar graphs (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof and show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then show that the partition function is independent of the orientations of the planar graphs so long as the orientations are Pfaffian. When these planar graphs are bipartite, we show that the computation of the partition function becomes especially simple. We then give an explicit product formula for the partition function of three-dimensional grid graphs a la Kasteleyn and Temperley--Fischer, which turns out to be fourth power of a polynomial when all grid lengths are even. Finally, we generalise this product formula to $d$ dimensions, again obtaining an explicit product formula. We conclude with a discussion on asymptotic formulas for the free energy and monopole densities.
