Hard Squares with Negative Activity and Rhombus Tilings of the Plane
Hard Squares with Negative Activity and Rhombus Tilings of the Plane
复制标题
具有负活动的硬方块和平面的菱形平铺
DOI：
--
复制
发表时间：
2006
影响因子：
0.7
通讯作者：
J. Jonsson
中科院分区：
文献类型：
--
作者：
J. Jonsson
Let $S_{m,n}$ be the graph on the vertex set ${Bbb Z}_m imes {Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,dpm 1)$ or $(a,b) = (c pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to ${Sigma_{m,n} : n ge 1}$. 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 $-1$.
Let $S_{m,n}$ be the graph on the vertex set ${Bbb Z}_m imes {Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,dpm 1)$ or $(a,b) = (c pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to ${Sigma_{m,n} : n ge 1}$. 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 $-1$.