论文标题
（随机）中间均匀生长的树木
(Random) Trees of Intermediate Uniform Growth
论文作者
论文摘要
对于每个足够良好的函数$ g：\ mathbb {r} _ {\ ge 0} \ rightarrow \ rightarrow \ mathbb {r} _ {\ ge 0} $，至少是线性生长的，至少是指数级的，我们最多可以构建均匀的体积增长$ g $，$ g $，$ c _1 c_1 cdot， | b_ {t}（v，r）| \ le c_2 \ cdot g（4r），\ quad \ text {对于所有$ r \ ge 0 $ and $ v \ in V（t）$}，$$，其中$ b_ {t}（t}（v，v，r）$表示Radius $ r $ r $ r $中心的Ball of Pertex $ v $ v $。特别是，这产生了均匀中间体树木（即超多项式和亚指数）体积生长的示例。 我们使用这种结构提供了一个均匀中间生长的单模型随机生根的树，回答了Itai Benjamini的一个问题。我们在成长中发现这些树的结构属性的变化$ r^{\ log \ log r} $。
For every sufficiently well-behaved function $g:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$ that grows at least linearly and at most exponentially we construct a tree $T$ of uniform volume growth $g$, that is, $$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 $r^{\log\log r}$.
