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A. J. Guttmann

Publications and source records attributed to A. J. Guttmann.

25 records · Page 2Linked to original sources
arXiv1999

Inversion relations, reciprocity and polyominoes

We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. For several classes of convex polygons, the inversion (reciprocity) relation, augmented by certain symmetry and analyticity properties, completely determines the anisotropic perimeter generating function.

math.CO↗
arXiv1998

Exact Enumeration Study of Free Energies of Interacting Polygons and Walks in Two Dimensions

We present analyses of substantially extended series for both interacting self-avoiding walks (ISAW) and polygons (ISAP) on the square lattice. We argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the theta-temperature, extending the application of a theorem of Tesi et. al. to two dimensions. Below the $θ$-temperature the conditions of this theorem break down, in contradistinction to three dimensions, but an analysis of the ratio of the partition functions for ISAP and ISAW indicates that the free energies are in fact equal at all temperatures to at least within 1%. Any perceived difference can be interpreted as the difference in the size of corrections-to-scaling in both problems. This may be used to explain the vastly different values of the cross-over exponent previously estimated for ISAP to that predicted theoretically, and numerically confirmed, for ISAW. An analysis of newly extended neighbour-avoiding SAW series is also given.

cond-mat.stat-mech↗
arXiv1998

On the Non-Universality of a Critical Exponent for Self-Avoiding Walks

We have extended the enumeration of self-avoiding walks on the Manhattan lattice from 28 to 53 steps and for self-avoiding polygons from 48 to 84 steps. Analysis of this data suggests that the walk generating function exponent gamma = 1.3385 +- 0.003, which is different from the corresponding exponent on the square, triangular and honeycomb lattices. This provides numerical support for an argument recently advanced by Cardy, to the effect that excluding walks with parallel nearest-neighbour steps should cause a change in the exponent gamma. The lattice topology of the Manhattan lattice precludes such parallel steps.

cond-mat.stat-mech↗
arXiv1998

Study of the Potts Model on the Honeycomb and Triangular Lattices: Low-Temperature Series and Partition Function Zeros

We present and analyze low-temperature series and complex-temperature partition function zeros for the $q$-state Potts model with $q=4$ on the honeycomb lattice and $q=3,4$ on the triangular lattice. A discussion is given as to how the locations of the singularities obtained from the series analysis correlate with the complex-temperature phase boundary. Extending our earlier work, we include a similar discussion for the Potts model with $q=3$ on the honeycomb lattice and with $q=3,4$ on the kagomé lattice.

cond-mat.stat-mech↗
arXiv1993

Critical exponents of the three-dimensional classical plane rotator model on the sc lattice from a high temperature series analysis

High temperature series expansions of the spin-spin correlation function for the plane rotator (or XY) model on the sc lattice are extended by three terms through order $β^{17}$. Tables of the expansion coefficients are reported for the correlation function spherical moments of order $l=0,1,2$. Our analysis of the series leads to fairly accurate estimates of the critical parameters.

hep-lat↗