(Random) Trees of Intermediate Uniform Growth
George Kontogeorgiou, Martin Winter

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 that grows at least linearly and at most exponentially we construct a tree of uniform volume growth , that is, where denotes the ball of radius centered at a vertex . 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 .
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Taxonomy
TopicsStochastic processes and statistical mechanics · Geometry and complex manifolds · Geometric Analysis and Curvature Flows
