On the independence complex of square grids
On the independence complex of square grids
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论方形网格的独立复形
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Eran Nevo
中科院分区：
文献类型：
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作者：
M. Bousquet;Svante Linusson;Eran Nevo
摘要
正则图的独立集合的计数在统计力学中很有意义，因为它对应于硬粒子模型的解。2004年，Fendley等人发现了它，对于某些复曲面边界条件的矩形网格，独立集的交错数是极其简单的。更准确地说，在矩形的边的互质条件下，偶数和奇数基数的独立集合的数量总是相差1。用物理学术语来说，这意味着在活度为-1的网格上观察硬粒子模型。这个猜想最近被琼森证明了。
在这里，我们产生了其他的网格图族，具有开放或圆柱形边界条件，对于它们，类似的性质在没有任何大小限制的情况下也成立：偶数和奇数基数的独立集的数量总是相差0，±1，或者在圆柱形的情况下，相差2的某个幂。
我们表明，这些结果反映了我们的图的独立复合体的更强的属性。我们使用Forman的离散莫尔斯理论确定这些复合物的同伦类型。我们发现，这些复合物要么是可收缩的，或同伦的一个领域，或者，在圆柱形的情况下，一个楔形的领域。
最后，我们使用我们的计数结果来确定在活度为-1时描述硬粒子模型的某些转移矩阵的谱。这些结果与Fendley等人的某些观点相一致，由Jonsson在Toric案例中证明。
Abstract
The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity −1. This conjecture was recently proved by Jonsson.
Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, ±1, or, in the cylindric case, by some power of 2.
We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman’s discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres.
Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity −1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case.