1,631 research outputs found

    Enumeration of self avoiding trails on a square lattice using a transfer matrix technique

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    We describe a new algebraic technique, utilising transfer matrices, for enumerating self-avoiding lattice trails on the square lattice. We have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding walk universality class. Assuming that trails behave like Aλnn1132A \lambda ^n n^{11 \over 32}, we find λ=2.72062±0.000006\lambda = 2.72062 \pm 0.000006 and A=1.272±0.002A = 1.272 \pm 0.002.Comment: To be published in J. Phys. A:Math Gen. Pages: 16 Format: RevTe

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

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    We present a new and more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A detailed comparison with the previous best algorithm shows significant improvement in the running time of the algorithm. The new algorithm is used to extend the enumeration of polygons to length 130 from the previous record of 110.Comment: 17 pages, 8 figures, IoP style file

    Comment on `Series expansions from the corner transfer matrix renormalization group method: the hard-squares model'

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    Earlier this year Chan extended the low-density series for the hard-squares partition function κ(z)\kappa(z) to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity zdz_d which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at zdz_d with exponents θ=0.83333(2)\theta=0.83333(2) and θ=1.6676(3)\theta'= 1.6676(3). We thus confirm that the exponent θ\theta has the exact value 56\frac56 as observed by Dhar.Comment: 5 pages, 1 figure, IoP macros. Expanded second and final versio