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

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 $A \lambda ^n n^{11 \over 32}$, we find $\lambda = 2.72062 \pm 0.000006$ and $A = 1.272 \pm 0.002$.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

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