TY - JOUR

T1 - On the independence complex of square grids

AU - Bousquet-Mélou, Mireille

AU - Linusson, Svante

AU - Nevo, Eran

N1 - Funding Information:
Acknowledgements We are grateful to Anders Björner and Richard Stanley for inviting us to the “Algebraic Combinatorics” program at the Institut Mittag-Leffler in Spring 2005, during which part of this work was done. All authors were partially supported by the European Commission’s IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”.

PY - 2008/6/1

Y1 - 2008/6/1

N2 - The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity -1. This conjecture was recently proved by Jonsson. Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, ±1, or, in the cylindric case, by some power of 2. We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman's discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres. Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity -1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case.

AB - The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity -1. This conjecture was recently proved by Jonsson. Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, ±1, or, in the cylindric case, by some power of 2. We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman's discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres. Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity -1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case.

KW - Discrete Morse theory

KW - Hard particles

KW - Independence complex

KW - Independent sets

KW - Transfer matrices

UR - http://www.scopus.com/inward/record.url?scp=42449110848&partnerID=8YFLogxK

U2 - 10.1007/s10801-007-0096-x

DO - 10.1007/s10801-007-0096-x

M3 - Article

AN - SCOPUS:42449110848

SN - 0925-9899

VL - 27

SP - 423

EP - 450

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 4

ER -