Hard Squares with Negative Activity on Cylinders with Odd Circumference
Hard Squares with Negative Activity on Cylinders with Odd Circumference
复制标题
奇数周长圆柱体上具有负活性的硬方块
DOI：
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发表时间：
2009
影响因子：
0.7
通讯作者：
J. Jonsson
中科院分区：
文献类型：
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作者：
J. Jonsson
设$C_{m，n}$是顶点集${1，ldots，m}上的图 当且仅当$（a，B）=（c，dpm 1，d）$或$（a，B）=（c，dpm 1，d）$时，其中在$（a，B）$和$（c，d）$之间存在边，其中第二索引以$n$为模计算。我们可以把$C_{m，n}$看作是一个周长为n$个单位的圆柱体上的单位正方形网格。对于奇数n，我们证明了C_{m，n}$中独立集的单纯复形$Sigma_{m，n}$的欧拉特征线是2 $或-1$，这取决于$gcd（m-1，n）$是否可被$3$整除。证明严重依赖于以前的工作，由于塔珀，谁减少了问题的计算欧拉特征的$Sigma_{m，n}$，以分析一定的子集有吸引力的性质。即使是$n$的情况仍然不清楚。在统计力学的语言中，$Sigma_{m，n}$的约化欧拉特征线与减去相应的活度为$-1$的硬平方模型的配分函数相吻合。
Let $C_{m,n}$ be the graph on the vertex set ${1, ldots, m} imes {0, ldots, n-1}$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,dpm 1)$ or $(a,b) = (c pm 1,d)$, where the second index is computed modulo $n$. One may view $C_{m,n}$ as a unit square grid on a cylinder with circumference $n$ units. For odd $n$, we prove that the Euler characteristic of the simplicial complex $Sigma_{m,n}$ of independent sets in $C_{m,n}$ is either $2$ or $-1$, depending on whether or not $gcd(m-1,n)$ is divisble by $3$. The proof relies heavily on previous work due to Thapper, who reduced the problem of computing the Euler characteristic of $Sigma_{m,n}$ to that of analyzing a certain subfamily of sets with attractive properties. The situation for even $n$ remains unclear. In the language of statistical mechanics, the reduced Euler characteristic of $Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.