Self-avoiding walks and trails on the 3.12 lattice

TL;DR

This paper derives the generating functions and growth constants for self-avoiding walks and trails on the $3.12^2$ lattice by relating them to known functions on the hexagonal lattice, and improves bounds on growth constants for related lattices.

Contribution

It provides the first explicit generating functions and growth constants for self-avoiding walks and trails on the $3.12^2$ lattice, using mappings to known lattice graphs.

Findings

Generated explicit formulas for self-avoiding walks and trails on the $3.12^2$ lattice.

Calculated growth constants based on known hexagonal lattice data.

Established improved bounds on growth constants for related lattices.

Abstract

We find the generating function of self-avoiding walks and trails on a semi-regular lattice called the $3.12^2$ lattice in terms of the generating functions of simple graphs, such as self-avoiding walks, polygons and tadpole graphs on the hexagonal lattice. Since the growth constant for these graphs is known on the hexagonal lattice we can find the growth constant for both walks and trails on the $3.12^2$ lattice. A result of Watson then allows us to find the generating function and growth constant of neighbour-avoiding walks on the covering lattice of the $3.12^2$ lattice which is tetra-valent. A mapping into walks on the covering lattice allows us to obtain improved bounds on the growth constant for a range of lattices.