Random walks on nilpotent groups driven by measures supported on powers of generators

TL;DR

This paper analyzes how convolution powers of specific measures supported on powers of generators decay on finitely generated nilpotent groups, revealing complex interactions between measure parameters and group structure.

Contribution

It characterizes the asymptotic decay of return probabilities for measures supported on powers of generators in nilpotent groups, extending understanding of random walks with heavy-tailed step distributions.

Findings

Decay rates depend on measure parameters and generator positions in the lower central series.

Behavior varies subtly with interactions between measure exponents and group structure.

Provides explicit asymptotics for return probabilities in this setting.

Abstract

We study the decay of convolution powers of a large family $\mu_{S,a}$ of measures on finitely generated nilpotent groups. Here, $S=(s_1,...,s_k)$ is a generating $k$-tuple of group elements and $a= (\alpha_1,...,\alpha_k)$ is a $k$-tuple of reals in the interval $(0,2)$. The symmetric measure $\mu_{S,a}$ is supported by $S^*=\{s_i^{m}, 1\le i\le k,\,m\in \mathbb Z\}$ and gives probability proportional to $$(1+m)^{-\alpha_i-1}$$ to $s_i^{\pm m}$, $i=1,...,k,$ $m\in \mathbb N$. We determine the behavior of the probability of return $\mu_{S,a}^{(n)}(e)$ as $n$ tends to infinity. This behavior depends in somewhat subtle ways on interactions between the $k$-tuple $a$ and the positions of the generators $s_i$ within the lower central series $G_{j}=[G_{j-1},G]$, $G_1=G$.