Existence and bounds of growth constants for restricted walks, surfaces, and generalisations

Abstract

We introduce classes of restricted walks, surfaces and their generalisations. For example, self-osculating walks (SOWs) are supersets of self-avoiding walks (SAWs) where edges are still not allowed to cross but may 'kiss' at a vertex. They are analogous to osculating polygons introduced in (Jensen and Guttmann, 1998) except that they are not required to be closed. The 'automata' method of (Pönitz and Tittmann, 2000) can be adapted to such restricted walks. For example, we prove upper bounds for the connective constant for SOWs on the square and triangular lattices to be $μ^{\mathrm{SOW}}_\square \leq 2.73911$ and $μ^{\mathrm{SOW}}_\triangle \leq 4.44931$, respectively. In analogy, we also introduce self-osculating surfaces (SOSs), a superset of self-avoiding surfaces (SASs) which can be generated from fixed polyominoids (XDs). We further generalise and define self-avoiding $k$-manifolds (SAMs) and its supersets, self-osculating $k$-manifolds (SOMs) in the $d$-dim hypercubic lattice and $(d, k)$-XDs. By adapting the concatenation procedure procedure (van Rensburg and Whittington, 1989), we prove that their growth constants exist, and prove an explicit form for their upper and lower bounds. The upper bounds can be improved by adapting the 'twig' method, originally developed for polyominoes (Eden, 1961, Klarner and Rivest, 1973). For the cubic lattice, we find improved upper bounds for the growth constant of SASs as $μ^{\mathrm{SAS}}_{\mathbb{Z}^3} \leq 17.11728$.