Iwan Jensen

Self-avoiding polygons on the square lattice

We have developed an improved algorithm that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 90. Analysis of the resulting series yields very accurate estimates of the connective constant =2.638 158 529 27(1) (biased) and the critical exponent = 0.500 0005(10) (unbiased). The critical point is indistinguishable from a root of the polynomial 581x4 + 7x2 - 13 = 0. An asymptotic expansion for the coefficients is given for all n. There is strong evidence for the absence of any non-analytic correction-to-scaling exponent.

Enumeration of self-avoiding walks on the square lattice

We describe a new algorithm for the enumeration of self-avoiding walks on the square lattice. Using up to 128 processors on a HP Alpha server cluster we have enumerated the number of self-avoiding walks on the square lattice to length 71. Series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and mean-square distance of monomers from the end points have been derived to length 59. An analysis of the resulting series yields accurate estimates of the critical exponents γ and ν confirming predictions of their exact values. Likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations. Finally we report on an analysis giving compelling evidence that the leading non-analytic correction-to-scaling exponent � 1 = 3/2.

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 < , which is much smaller than the growth factor = 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.

A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice

We have developed a parallel algorithm that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 110. We have also extended the series for the first 10 area-weighted moments and the radius of gyration to 100. Analysis of the resulting series yields very accurate estimates of the connective constant ? = 2.638 158 530 31(3) (biased) and the critical exponent ? = 0.500 000 1(2) (unbiased). In addition, we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations.

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, λ=4.062570(8), for the growth constant of lattice animals, and, λ0=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, ν=0.64115(5), for the size exponent.

Counting polyominoes: a parallel implementation for cluster computing

The exact enumeration of most interesting combinatorial problems has exponential computational complexity. The finite-lattice method reduces this complexity for most two-dimensional problems. The basic idea is to enumerate the problem on small finite lattices using a transfer-matrix formalism. Systematically grow the size of the lattices and combine the results in order to obtain the desired series for the infinite lattice limit. We have developed a parallel algorithm for the enumeration of polyominoes, which are connected sets of lattice cells joined at an edge. The algorithm implements the finite-lattice method and associated transfer-matrix calculations in a very efficient parallel setup. Test runs of the algorithm on a HP server cluster indicates that in this environment the algorithm scales perfectly from 2 to 64 processors.

Scaling function and universal amplitude combinations for self-avoiding polygons

We analyse new data for self-avoiding polygons (SAPs), on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal amplitude combinations for all area moments and suggest that rooted SAPs may satisfy a q-algebraic functional equation.

Critical exponents for branching annihilating random walks with an even number of offspring.

Recently, Takayasu and Tretyakov [Phys. Rev. Lett. 68, 3060 (1992)] studied branching annihilating random walks (BAWs) with n=1\char21{}5 offspring. These models exhibit a continuous phase transition to an absorbing state. For odd n the models belong to the universality class of directed percolation. For even n the particle number is conserved modulo 2 and the critical behavior is not compatible with directed percolation. In this article I study the BAW with n=4 using time-dependent simulations and finite-size scaling, obtaining precise estimates for various critical exponents. The results are consistent with the conjecture: \ensuremath{\beta}/

A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice

{\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}

Low-density series expansions for directed percolation on square and triangular lattices

=1/2,