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 found from square lattice series.