(Random) Trees of Intermediate Uniform Growth

TL;DR

This paper constructs trees with a wide range of intermediate uniform growth rates, including super-polynomial and sub-exponential, and explores their structural properties, answering a question about unimodular random trees.

Contribution

It introduces a method to construct trees with prescribed intermediate growth rates and provides the first examples of unimodular random trees with such growth, revealing structural changes at specific growth thresholds.

Findings

Constructed trees with growth between polynomial and exponential

Provided examples of unimodular random trees with intermediate growth

Identified structural property changes at growth rate r^{log log r}

Abstract

For every sufficiently well-behaved function $g:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$ that grows at least linearly and at most exponentially we construct a tree $T$ of uniform volume growth $g$, that is, $$C_1\cdot g(r/4)\le |B_{T}(v,r)| \le C_2\cdot g(4r),\quad\text{for all $r\ge 0$ and $v\in V(T)$},$$ where $B_{T}(v,r)$ denotes the ball of radius $r$ centered at a vertex $v$. In particular, this yields examples of trees of uniform intermediate (i.e. super-polynomial and sub-exponential) volume growth. We use this construction to provide first examples of unimodular random rooted trees of uniform intermediate growth, answering a question by Itai Benjamini. We find a peculiar change in structural properties for these trees at growth $r^{\log\log r}$.