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{\bf Jakob Jonsson}
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{\bf Hard Squares with Negative Activity and Rhombus Tilings of the Plane}
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Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times
{\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if
and only if either $(a,b) = (c,d\pm 1)$ 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 $-1$.
\bye