Hard squares with negative activity and rhombus tilings of the plane

• Let S-m,S-n be the graph on the vertex set Z(m) x Z(n) in which there is an edge between (a, b) and (c, d) if and only if either (a, b) = (c, d +/- 1) or (a, b) (c +/- 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,S-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(x(I) - T-m) are roots of unity, where T-m is a certain transfer matrix associated to {Sigma(m,n) : n >= 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.

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