On counting polygons in a crystal

TL;DR

This paper extends the understanding of polygon counting in infinite quasi-transitive graphs with specific symmetry properties, identifying the exponential growth rate as the connective constant, generalizing classical lattice results.

Contribution

It generalizes Hammersley's result on hypercubic lattices to a broader class of graphs with certain automorphism actions, addressing a longstanding challenge.

Findings

Identifies the exponential growth rate as the connective constant for polygons in these graphs.

Extends classical lattice results to more general 'crystal' graphs with specific symmetries.

Provides a framework for counting polygons in graphs with a ${b Z}^2$ automorphism action.

Abstract

How many $n$-step polygons exist that contain a given vertex of an infinite quasi-transitive graph $G$? The exponential growth rate of such polygons is identified as the connective constant when $G$ has sub-exponential growth and possesses a so-called square graph height function. The last condition amounts to the requirement that $G$ has a certain ${\Bbb Z}^2$ action of automorphisms. The main theorem extends a result of Hammersley (Proc. Cambridge Philos. Soc. 57 (1961) 516--523) and others for the hypercubic lattice, and responds to Hammersley's challenge to prove such a result for more general "crystals''.