Connective Constants on Nested Fractal Graphs

Abstract

We study self-avoiding walks on the canonical one-sided graphs of Lindstrom nested fractals. We prove that the connective constant $\mu$ exists and identify $\log \mu$ with the critical inverse temperature of a finite-dimensional boundary-state renormalization. If the boundary-state partition vectors are bounded at criticality, then the fixed-length counts $c_n$ satisfy two-sided polynomial bounds around ${\mu }^n$. We also prove that $h$-flexibility implies $c_{n+h}/c_n\to {\mu }^h$. For regular polygonal $N$-gaskets, we derive exact crossing recursions, determine the smallest flexibility step $h$, and obtain explicit algebraic connective constants for the $6$- and $9$-gaskets. The Vicsek graph has no flexibility step, and its successive ratios do not converge.

50 pages, 8 figures