Algebraic languages and polyominoes enumeration

126

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. 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.

Section: The Metric Propertiesmentioning

confidence: 60%

Enumeration of self-avoiding walks on the square lattice

Jensen¹

2004

126

“…For a cluster which terminates after t growth stages (and therefore has length L = t + 1) the walks will meet after each one has executed t + 1 steps. The number of pairs of walks of the above type with a given length has been enumerated by Delest and Viennot (1984) and their result may be taken over to give the following formula for the number of compact clusters w, having exactly t growth stages:…”

Section: Isotropic Clusters Grown From a Seed Of Width Onementioning

confidence: 99%

Directed compact percolation: cluster size and hyperscaling

Essam

1989

Abstract. Exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice. The corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained. The work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice. A duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve. The moments of both distributions have critical exponents which satisfy a c o t " gap hypothesis with gap exponents U,-, = 2 and A = 3 corresponding to the scaling length and scaling size respectively. The usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced bywhere U, is the exponent corresponding to the cluster width and D is the number of transverse dimensions ( = 1 for the square lattice).

“…Secondly, as we look through the configuration starting from the bottom the number of '0's is never smaller than the number of '1's. Those familiar with algebraic languages will immediately recognise that each configuration of labelled loop-ends forms a Dyck word (see [25]). …”

Section: Enumeration Of Closed Connected Meandersmentioning

confidence: 99%

A transfer matrix approach to the enumeration of plane meanders

Jensen

2000

A closed plane meander of order n is a closed self-avoiding curve intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on transfer matrix methods, for the enumeration of plane meanders. While the algorithm has exponential complexity, its rate of growth is much smaller than that of previous algorithms. The algorithm is easily modified to enumerate various systems of closed meanders, semi-meanders, open meanders and many other geometries.