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Percolation on two-dimensional small-world networks has been proposed as a model for the spread of plant diseases. In this paper we give an analytic solution of this model using a combination of generating function methods and high-order series expansion. Our solution gives accurate predictions for quantities such as the position of the percolation(More)
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(More)
  • Iwan Jensen
  • 2003
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(More)
We calculate improved lower bounds for the connective constants for self-avoiding walks on the square, hexagonal, triangular, (4.8 2), and (3.12 2) lattices. The bound is found by Kesten's method of irreducible bridges. This involves using transfer-matrix techniques to exactly enumerate the number of bridges of a given span to very many steps. Upper bounds(More)
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 λ < 8 √ 2, which is much smaller than the growth factor λ = 4 √ 2 of the previous best(More)