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On counting polygons in a crystal

Geoffrey R. Grimmett

TL;DR

The work extends Hammersley’s classical relation between self-avoiding walks and polygons from ${\mathbb Z}^d$ to a broad class of infinite quasi-transitive graphs that admit a square graph height function. By introducing and exploiting square ghfs, the authors establish that the polygon growth rate satisfies $\pi(G)=\mu(G)=\beta(G)$ for graphs with sub-exponential growth, with explicit construction via $h$-bridges and a tubular region that assembles long polygons. The results are illustrated on Euclidean lattices and contrast with examples (e.g., certain trees and ladders) where the square ghf fails, demonstrating sharpness of the conditions. The paper connects to prior work on connective constants, discusses implications for ballisticity in non-amenable settings, and provides a framework that broadens Hammersley’s crystal challenge beyond ${\mathbb Z}^d$. Overall, it contributes a rigorous criterion for the equality of growth rates of SAWs and polygons in a wide array of crystal-like graphs.

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''.

On counting polygons in a crystal

TL;DR

The work extends Hammersley’s classical relation between self-avoiding walks and polygons from to a broad class of infinite quasi-transitive graphs that admit a square graph height function. By introducing and exploiting square ghfs, the authors establish that the polygon growth rate satisfies for graphs with sub-exponential growth, with explicit construction via -bridges and a tubular region that assembles long polygons. The results are illustrated on Euclidean lattices and contrast with examples (e.g., certain trees and ladders) where the square ghf fails, demonstrating sharpness of the conditions. The paper connects to prior work on connective constants, discusses implications for ballisticity in non-amenable settings, and provides a framework that broadens Hammersley’s crystal challenge beyond . Overall, it contributes a rigorous criterion for the equality of growth rates of SAWs and polygons in a wide array of crystal-like graphs.

Abstract

How many -step polygons exist that contain a given vertex of an infinite quasi-transitive graph ? The exponential growth rate of such polygons is identified as the connective constant when has sub-exponential growth and possesses a so-called square graph height function. The last condition amounts to the requirement that has a certain 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''.
Paper Structure (10 sections, 6 theorems, 33 equations, 5 figures)

This paper contains 10 sections, 6 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main theorem
  4. Statement of main theorem
  5. Euclidean lattices
  6. Tree graphs
  7. Ladder graphs
  8. Products of trees and lines
  9. Comments on main theorem
  10. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $G\in{\mathcal{G}}$ have a square graph height function and sub-exponential growth, and let $p_n$ be the number of $n$-step polygons that include the root of $G$. Then $\pi(G) :=\limsup_{n\to \infty}p_n^{1/n}$ satisfies $\pi(G) = \mu(G)$.

Figures (5)

  • Figure 3.1: The hexagonal lattice and the square/octagon lattice. The heights of vertices are as marked. The automorphism $\rho$ is a suitable shift rightwards for the first, and a suitable shift upwards for the second.
  • Figure 3.2: The $3$-regular tree with the ' horocyclic' height function.
  • Figure 3.3: The ladder graph ${\mathbb L}_2$ with the root and heights indicated.
  • Figure 4.1: The region $R_n$. A shortest path $l_n$ joins ${\bf 0}$ to $P_n'$, and the red (dashed) path is a bridge. The two vertical lines depict the sets of $v$ such that $h(v)=0$, and such that $h(v)=h(P_n')$, respectively.
  • Figure 4.2: The tube $S_N$ and its image $\rho^k(S_N)$. Each is traversed by an $(n+\ell)N$-step bridge, and these two bridges are joined into a red (dashed) polygon by adding the connecting paths $\nu^\pm$.

Theorems & Definitions (13)

  • Theorem 1.1: Equality of growth rates
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • ...and 3 more