Numerical Study of a Conjecture in the Self-avoiding Random Walk Problem

Abstract

It has been conjectured that the sum of the critical attritions p. and v of a selfavoiding random walk on a triangular and a honeycomb lattice respectively should be precisely six. Estimates of the critical attrition obtained from the analysis of exact series expansions support this conjecture. Assuming the conjecture, estimates of the two critical attritions are made and found to be in good agreement with those obtained by other methods. The exact inequality v2 ~ p.2/(1 + p.) is proved, and it is shown that an analogous inequality applies to a pair of three-dimensional lattices.