Papers

Abstract: The 6-cube has a total of 7174574164703330195841 matchings of which 16332454526976 are perfect. This was computed with a transfer matrix method associated with polygraphs. For polygraphs of type G x P_m we present a method for compression of the transfer matrix. This compression gives a substantial reduction of the order of transfer matrix by exploiting the automorphisms of the graph G. We compute and tabulate matching polynomials of various polygraphs, such as the 4 x 4 x m grid. A Mathematica package, GrafPack, is demonstrated and used for computation of matching polynomials, permanents and for generating transfer matrices.

Abstract: We present a method for reducing the size of transfer matrices by exploiting symmetry. For example, the transfer matrix for enumeration of matchings in the graph C₄ x C₄ x P_n can be reduced from order 65536 to 402 simply due to the 384 automorphisms of C₄ x C₄. The matrix for enumeration of perfect matchings can be still further reduced to order 93, all in a straightforward and mechanical way. As an application we report an improved upper bound for the 3-dimensional dimer problem.

Abstract: The Ising partition function for a graph counts the number of bipartitions of the vertices into sets of given sizes, with a given size of the induced edge cut. This is expressed as a 2 variable generating function which is easily translatable into the partition function studied in statistical physics. The author has exactly computed this generating function for the square-, triangular- and union jack grid with open boundary, the largest having 256 vertices. We describe the computing process, which is based upon transfer matrices and a simple use of symmetry, and do a rudimentary analysis of the phase transition which occurs when the edge cut attains a certain critical size.

Abstract: The Ising partition function for a graph counts the number of bipartitions of the vertices with given sizes, with a given size of the induced edge cut. Expressed as a 2-variable generating function it is easily translatable into the corresponding partition function studied in statistical physics. In the current paper a comparatively efficient transfer matrix method is described for computing the generating function for the n x n grid with periodic boundary. We have applied the method to up to the 15 x 15 grid, in total 225 vertices. We examine the phase transition that takes place when the edge cut reaches a certain critical size. From the physical partition function we extract quantities such as magnetisation and susceptibility and study their asymptotic behaviour at the critical temperature. The partition function in one variable is considerably easier to compute for large grids. We study this partition function for the 128 x 128 grid.

Abstract: Monte Carlo simulation of the Ising model is often used with a single spin flip strategy such as the Metropolis method. We conjecture that for the 3-dimensional model using the Metropolis condition, the probability that a randomly selected vertex is flipped is 1/2 at the critical temperature in the thermodynamical limit. Using the Glauber condition we conjecture this number to be 1/3. Estimates of these probabilities are given for cubes of linear order up to 2048 for three different couplings.