Enumerations of lattice animals and trees

We have developed an improved algorithm that allows us to enumerate the number of site animals on the square lattice up to size 46. We also calculate the number of lattice trees up to size 44 and the radius of gyration of both lattice animals and trees up to size 42. Analysis of the resulting series yields an improved estimate, $\lambda = 4.062570(8)$, for the growth constant of lattice animals, and, $\lambda_0 = 3.795254(8)$, for the growth constant of trees, and confirms to a very high degree of certainty that both the animal and tree generating functions have a logarithmic divergence. Analysis of the radius of gyration series yields the estimate, $\nu = 0.64115(5)$, for the size exponent.Comment: 14 pages, 2 eps figures, corrections to some series coefficients and reference

Low-density series expansions for directed percolation I: A new efficient algorithm with applications to the square lattice

A new algorithm for the derivation of low-density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems. Numerical evidence shows that the computational complexity grows exponentially, but with a growth factor \lambda < \protect{\sqrt[8]{2}}, which is much smaller than the growth factor \lambda = \protect{\sqrt[4]{2}} of the previous best algorithm. For bond (site) percolation on the directed square lattice the series has been extended to order 171 (158). Analysis of the series yields sharper estimates of the critical points and exponents.Comment: 20 pages, 8 figures (3 of them > 1Mb

Fuchsian differential equation for the perimeter generating function of three-choice polygons

Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of three-choice polygons. We find that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We perform an analysis of the properties of the differential equation.Comment: 13 pages, 2 figures, talk presented in honour of X. Viennot at Seminaire Lotharengien, Lucelle, France, April 3-6 2005. Paper amended and sligtly expanded after refereein