Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Hard Squares with Negative Activity and Rhombus Tilings of the Plane. | Hard Squares with Negative Activity and Rhombus Tilings of the Plane Jakob Jonsson Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139 jakob@ Submitted Mar 24 2006 Accepted Jul 28 2006 Published Aug 7 2006 Mathematics Subject Classifications 05A15 05C69 52C20 Abstract Let Sm n be the graph on the vertex set Zm X Zn 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 of independent sets in Sm n. In particular we show that the unreduced Euler characteristic of ym 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 ym 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 I Tm are roots of unity where Tm is a certain transfer matrix associated to ym n n 1 . In the language of statistical mechanics the reduced Euler characteristic of ym n coincides with minus the partition function of the corresponding hard square model with activity 1. 1 Introduction An independent set in a simple and loopless graph G is a subset of the vertex set of G with the property that no two vertices in the subset are adjacent. The family of independent sets in G forms a simplicial complex the independence complex E G of G. The purpose of this paper is to analyze the independence complex of square grids with periodic boundary conditions. Specifically define Sm n to be the graph with vertex set Zm X Zn and with an edge between a b and c d if and only if either a b c d 1 or a b c 1 d computations carried out modulo m n . Defining Lm n mZ X nZ Research supported by the European Graduate Program Combinatorics Geometry and Computation DFG-GRK 588 2. THE ELECTRONIC JOURNAL OF .
