September 3, 2008
Jakob Jonsson (KTH): Hard squares with negative activity on cylinders with odd circumference
Abstract:
Let G_{m,n} be the graph on the vertex set {1, ..., m} \times {1, ..., 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), where the second
index is computed modulo n. One may view G_{m,n} as a unit square grid
on a cylinder with circumference n and height m. For odd n, we prove
that the Euler characteristic of the simplicial complex \Sigma_{m,n} of
independent sets in G_{m,n} is either 2 or -1, depending on whether
or not gcd(m-1,n) is divisible by 3. The proof builds on
previous work due to Johan 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.