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 Gω of a general Grigorchuk group encoded by a sequence ω∈{0,1,2}Bbb N. In particular, μ(Gω) > ϕ for any such Cayley graph (subject to a simple condition on ω), where ϕ:= 1/2(1+√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)≥ϕ 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.
Related
- The growth of Grigorchuk's torsion group
- The twisted twin of the Grigorchuk group
- Commutator width in the first Grigorchuk group
- On the residual finiteness of the non-abelian exterior square of wreath products and the Grigorchuk group
- Motivic Segre classes of Schubert cells and the connective formal group law
BibTeX
@misc{grimmett2026connective,
title = {{Connective constants of Grigorchuk graphs}},
author = {Geoffrey R. Grimmett},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.19349},
eprint = {2608.19349},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19349},
}
RIS
TY - UNPB
TI - Connective constants of Grigorchuk graphs
AU - Geoffrey R. Grimmett
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.19349
UR - https://arxiv.org/abs/2608.19349
AB - 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∈{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:= 1/2(1+√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)≥ϕ 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.
ER -