AI 中文总结
该研究针对Lindstrom嵌套分形的正则单侧图，证明其自回避游走的连通常数μ存在，推导正多边形N- gasket的连通常数递推式，明确Vicsek图无灵活步长且相邻比值不收敛的特性。
AI 中文摘要
我们研究Lindstrom嵌套分形的正则单侧图上的自回避游走，证明连通常数μ存在，并将logμ与有限维边界态重整化的临界逆温度对应。若临界时边界态配分向量有界，则固定长度计数c_n在μ^n附近满足双侧多项式界；还证明h-灵活性蕴含c_{n+h}/c_n→μ^h。对正多边形N- gasket，推导精确穿越递推式、确定最小灵活步长h，并得到6- gaskets与9- gaskets的显式代数连通常数。Vicsek图无灵活步长，其相邻比值不收敛。
英文摘要
We study self-avoiding walks on the canonical one-sided graphs of Lindstrom nested fractals. We prove that the connective constant $μ$ exists and identify $\logμ$ 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 $μ^n$. We also prove that $h$-flexibility implies $c_{n+h}/c_n\toμ^h$. 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.
Comments50 pages, 8 figures
