A Unimodular Random Graph With Large Upper Growth and No Growth
Abstract
ABSTRACT We construct a unimodular random rooted graph with maximal degree and upper growth rate , which does not have a growth rate. Abért, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least , then the growth rate exists, and asked with some scepticism if this may hold for more general graphs. Our construction shows that the answer is negative. We also provide a non‐hyperfinite example of a unimodular random graph with no growth rate. This may be of interest in light of a conjecture of Abért that unimodular random Riemannian surfaces of bounded negative curvature always have growth.
Fields of Study & Subjects
Advanced Graph Theory Research
Stochastic processes and statistical mechanics
Cooperative Communication and Network Coding
Publication details
- Journal / source
- Random Structures and Algorithms
- Volume
- 68
- Issue
- 4
- Publication date
- 29 May 2026
- Language
- EN