Self-avoiding polygons on the square, L and Manhattan lattices

Abstract

Transfer-matrix techniques are used to extend the self-avoiding polygon generating function on the square lattice to terms in x46, corresponding to 46 step polygons. These techniques are then extended to apply to directed square lattices, such as the L and Manhattan lattice, and the self-avoiding polygon generating function to x48 is found for these lattices. Series analysis confirms that the 'specific heat' exponent alpha =1/2 for the self-avoiding walk problem, and gives the following estimates for the connective constants: mu (SQ)=2.638155+or-0.000025, mu (L)=1.5657+or-0.0019 and mu (Man.)=1.7328+or-0.0005. Some evidence for a correction to scaling exponent Delta approximately=0.84 is fou..