Work

A Unimodular Random Graph With Large Upper Growth and No Growth

Year: 2026

Type: article

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.

Source: Random Structures and Algorithms

DOI: 10.1002/rsa.70079

Authors: Péter Mester, Ádám Timár

Institutions: University of Iceland, HUN-REN Alfréd Rényi Institute of Mathematics

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: H2020 European Research Council

Award: G1472956110