Self-avoiding Walks of Lattice Strips

TL;DR

This paper investigates self-avoiding walks on specific lattice strips, providing new methods to compute the connective constant for one strip and bounds for another, along with estimates for walk counts.

Contribution

It introduces a novel shorter approach to determine the connective constant for the old lattice strip and establishes bounds for both studied strips.

Findings

Connective constant for old strip computed with a new method

Bounds for the connective constant of the old strip established

Lower and upper bounds for the number of SAWs of length n derived

Abstract

We study self-avoiding walks on restricted square lattices, more precisely on the lattice strips $\mathbb{Z} \times \{-1,0,1\}$ and $\mathbb{Z}\times \{-1,0,1,2\}$. We obtain the value of the connective constant for the $\mathbb{Z} \times \{-1,0,1\}$ lattice in a new shorter way and deduce close bounds for the connective constant for the $\mathbb{Z}\times \{-1,0,1,2\}$ lattice. Moreover, for both lattice strips we find close lower and upper bounds for the number of SAWs of length $n$ by using the connective constant.