Study on dimer models containing impurities

平面二部图上的完美匹配作为二聚体模型在统计力学中得到了广泛的研究。对于底层图是非二部的情况，几乎没有已知的结果。本文研究了平面二部图（如正方形图和六边形图）的完美匹配问题。连接最近的黑色（或白色）顶点的额外边。因此，这种图上的完美匹配在某种意义上可以认为是含有杂质的完美匹配。我们将自己的结果推广到只有一个杂质的模型中，得到了多个杂质存在情况下的一些结果。但这些结果是有限的，当比较单一的杂质情况。

Perfect matchings on planar bipartite graphs have been extensively studied as the dimer models in statistical mechanics. Almost no results are known for the case in which the underlying graphs are non-bipartite. In this study, we investigate the perfect matchings on the graphs which can be obtained by adding some extra edges tothe planar bipartite graphs, such as square graphs and hexagonal graphs. Extra edges connecting nearest black(or white) vertices. Therefore the perfect matchings on such graphs can be considered as perfect matchings containing impurities in some sence.Generalizing our own results which describes the models with only one, we obtained some results on the cases in which multiple impurities exist. But these results are limited when compared to the single impurity case.