Connective constants of Grigorchuk graphs

Abstract: The connective constant $μ(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We prove upper and lower bounds for the connective constants of Cayley graphs $G_ω$ of a general Grigorchuk group encoded by a sequence $ω\in{0,1,2}^{\Bbb N}$. In particular, $μ(G_ω) > φ$ for any such Cayley graph (subject to a simple condition on $ω$), where $φ:= \frac12(1+\sqrt 5)$ is the golden mean. This extends earlier work of the author and Zhongyang Li in "Cubic graphs and the golden mean'', Discrete Math. 343 (2020), article 111638, where it was conjectured that $μ(G)\geφ$ for all infinite, vertex-transitive, cubic graphs. The current work includes an analysis of the proportions of appearances of given label-sequences in the orbital Schreier graphs of general Grigorchuk groups.

• The paper proves that for every Grigorchuk group encoding $\omega \in \{0,1,2\}^{\mathbb{N}$, the connective constant $\mu(G_\omega(\neg z))$ of the Cayley graph $G_\omega(\neg z)$ is at least the golden mean $\phi = \frac{1}{2}(1 + \sqrt{5})$.
• Explicit numerical bounds for the connective constants are provided using generating functions, with strict improvements under certain conditions on $\omega$.
• The methodology involves Schreier graph combinatorics, validating that certain self-avoiding walks (SAWs) on the Cayley graph correspond to walks on the Schreier graph without returning to specific structures more than once.

Overview and main result

This paper, by Geoffrey R. Grimmett, studies the connective constant $\mu(G)$ — the exponential growth rate of the number $c_n$ of $n$-step self-avoiding walks (SAWs) from a root, proved to exist by Hammersley — for Cayley graphs of general Grigorchuk groups. Each such group $\Gamma_\omega$ is encoded by an infinite vector $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$, with the classical first Grigorchuk group corresponding to $\omega = (012)^{\mathbb{N}}$. The paper extends earlier work of Grimmett and Li [(2608.19349)'s predecessor, GL20], which conjectured that every infinite, vertex-transitive, cubic graph has connective constant at least the golden mean $\phi = \frac{1}{2}(1+\sqrt{5})$.

The central result affirms this conjecture for all three-generator Cayley graphs of general Grigorchuk groups:

• Theorem (main): For every $\omega \in \{0,1,2\}^{\mathbb{N}}$, each cubic Cayley graph $G_\omega(\neg z)$ satisfies $\mu(G_\omega(\neg z)) \ge \phi$. Moreover, the inequality is strict whenever $c_n$0, i.e., whenever $c_n$1 contains all three values of $c_n$2.

This is a strict strengthening of the earlier result for the first Grigorchuk group in GL20, which established only the weak inequality. The strictness claim is notable: it shows the golden mean is not attained by these graphs, consistent with the ladder graph being the unique extremal candidate among known examples.

Quantitative bounds

The paper provides explicit numerical two-sided bounds via generating-function arguments:

Graph	Lower bound	Upper bound
$c_n$3 (4 generators)	$c_n$4	$c_n$5
$c_n$6 where $c_n$7	$c_n$8	$c_n$9
$n$0, $n$1	$n$2 (strict under condition on $n$3)	$n$4

Here $n$5 are reciprocals of positive roots of explicit polynomial equations. For example, $n$6 is the reciprocal of the root of $n$7, and $n$8 that of $n$9. The upper bound for the degree-4 graph exploits that $\Gamma_\omega$0 is dominated by the free-product graph $\Gamma_\omega$1 obtained by discarding relators, whose connective constant was computed by Gilch; strict inequality follows from the comparison theorem of Grimmett–Li.

A corollary of these upper bounds, via the standard inequality $\Gamma_\omega$2, is an improved lower bound on bond-percolation thresholds $\Gamma_\omega$3, complementing the result of Muchnik and Pak that $\Gamma_\omega$4 for all $\Gamma_\omega$5.

Method: Schreier graph combinatorics

The proof architecture rests on the orbital Schreier graph $\Gamma_\omega$6 of the ray $\Gamma_\omega$7 in the rooted binary tree $\Gamma_\omega$8. This graph is a singly infinite chain with parallel edges carrying labels from the generator set; between consecutive pairs of $\Gamma_\omega$9-edges there are two loops and two non-loops, with loop-labels determined by the encoding $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$0. Walks on $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$1 that move rightward or traverse loops lift to genuine SAWs on the Cayley graph $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$2: a key lemma shows that no such lifted walk can return to a "kite" (a translate of the Klein 4-group, appearing as a copy of $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$3) more than once, since that would force a forbidden cycle in the Schreier graph.

The counting proceeds by decomposing walks into units between renewal points (endpoints of $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$4-edges), yielding generating functions bounded below by geometric sums of unit polynomials. A refinement replaces short crossing words in certain units by longer detours (e.g., substituting $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$5 for $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$6), enlarging the counted SAW family and strictly improving the resulting lower bound. For case (c), the golden-mean bound is obtained by verifying that both unit polynomials evaluated at $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$7 equal or exceed $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$8.

Label-sequence densities

An independent contribution is a direct analysis of label frequencies in $$\omega = (\omega_0, \omega_1, \dots) \in \{0,1,2\}^{\mathbb{N}}$$9. Writing $\omega = (012)^{\mathbb{N}}$0 for the rays associated with vertices $\omega = (012)^{\mathbb{N}}$1, the paper proves:

• Theorem: Every word $\omega = (012)^{\mathbb{N}}$2 is periodic with period $\omega = (012)^{\mathbb{N}}$3 in the sequence $\omega = (012)^{\mathbb{N}}$4, and consequently the asymptotic frequency $\omega = (012)^{\mathbb{N}}$5 exists. Equivalently, the empirical distribution of $\omega = (012)^{\mathbb{N}}$6 converges weakly to the uniform measure on boundary rays beginning with $\omega = (012)^{\mathbb{N}}$7.
• Corollary: The frequency $\omega = (012)^{\mathbb{N}}$8 of generator $\omega = (012)^{\mathbb{N}}$9 as a loop-label at vertices $\phi = \frac{1}{2}(1+\sqrt{5})$0 equals $\phi = \frac{1}{2}(1+\sqrt{5})$1, explicitly computable from $\phi = \frac{1}{2}(1+\sqrt{5})$2.

The proof is by an elementary induction on nested word-evolution tables rather than via symbolic dynamics; the author notes that comparable results likely follow from the subshift machinery of Grigorchuk, Lenz, Nagnibeda and others. These densities feed directly into the improved strict bound: when some $\phi = \frac{1}{2}(1+\sqrt{5})$3 (the deleted generator's identity entry occurs), the proportion $\phi = \frac{1}{2}(1+\sqrt{5})$4 of augmented units is strictly positive, and the lower bound becomes the root of $\phi = \frac{1}{2}(1+\sqrt{5})$5, which lies strictly above $\phi = \frac{1}{2}(1+\sqrt{5})$6.

Limitations and open questions

Several caveats qualify the results. The general Question — whether $\phi = \frac{1}{2}(1+\sqrt{5})$7 for all infinite transitive cubic graphs — remains open; this paper resolves it only within the Grigorchuk family. The strict inequality in case (c) requires the condition that some entry $\phi = \frac{1}{2}(1+\sqrt{5})$8 assigns the identity to the deleted generator; sequences failing this condition receive only the weak bound $\phi = \frac{1}{2}(1+\sqrt{5})$9, and the paper does not determine whether equality can occur there. The numerical bounds depend primarily on $\omega \in \{0,1,2\}^{\mathbb{N}}$0, so the estimates are coarse for many encodings. Additionally, Grigorchuk groups admit no graph height function (per GL and Antoniuk–Biswas), which rules out several standard techniques and motivates the bespoke Schreier-graph approach used here.

Conclusion

The paper establishes $\omega \in \{0,1,2\}^{\mathbb{N}}$1, with strictness under a simple condition on the encoding sequence, for all cubic Cayley graphs of general Grigorchuk groups, together with explicit quantitative bounds and a self-contained analysis of label frequencies in their orbital Schreier graphs. The techniques — unit decomposition, lifting from Schreier graphs, and kite-based non-return arguments — provide a template that may apply to other branch groups, though the general cubic-transitive question remains unresolved.