(Random) Trees of Intermediate Uniform Growth

For every sufficiently well-behaved function $g:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ that grows at least linearly and at most exponentially we construct a tree $T$ of uniform volume growth $g$, that is, $$C_1\cdot g(r\mathrm{/}4)\le \mathrm{\mid }{B}_T(v,r)\mathrm{\mid }\le C_2\cdot g(4r),\text{for all }r\ge 0\text{ and }v\in V(T),$$ 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 $r^{\log \log r}$.