1,817 research outputs found
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. Analysis of the resulting series
yields accurate estimates of the critical exponents and
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 .Comment: 24 pages, 6 figure
Circuits in random graphs: from local trees to global loops
We compute the number of circuits and of loops with multiple crossings in
random regular graphs. We discuss the importance of this issue for the validity
of the cavity approach. On the one side we obtain analytic results for the
infinite volume limit in agreement with existing exact results. On the other
side we implement a counting algorithm, enumerate circuits at finite N and draw
some general conclusions about the finite N behavior of the circuits.Comment: submitted to JSTA
Three-dimensional maps and subgroup growth
In this paper we derive a generating series for the number of cellular
complexes known as pavings or three-dimensional maps, on darts, thus
solving an analogue of Tutte's problem in dimension three.
The generating series we derive also counts free subgroups of index in
via a simple bijection
between pavings and finite index subgroups which can be deduced from the action
of on the cosets of a given subgroup. We then show that this
generating series is non-holonomic. Furthermore, we provide and study the
generating series for isomorphism classes of pavings, which correspond to
conjugacy classes of free subgroups of finite index in .
Computational experiments performed with software designed by the authors
provide some statistics about the topology and combinatorics of pavings on
darts.Comment: 17 pages, 6 figures, 1 table; computational experiments added; a new
set of author
Self-avoiding walks and polygons on the triangular lattice
We use new algorithms, based on the finite lattice method of series
expansion, to extend the enumeration of self-avoiding walks and polygons on the
triangular lattice to length 40 and 60, respectively. For self-avoiding walks
to length 40 we also calculate series for the metric properties of mean-square
end-to-end distance, mean-square radius of gyration and the mean-square
distance of a monomer from the end points. For self-avoiding polygons to length
58 we calculate series for the mean-square radius of gyration and the first 10
moments of the area. Analysis of the series yields accurate estimates for the
connective constant of triangular self-avoiding walks, ,
and confirms to a high degree of accuracy several theoretical predictions for
universal critical exponents and amplitude combinations.Comment: 24 pages, 6 figure
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