# Groups, Geometry, and Dynamics

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**Volume 9, Issue 4, 2015, pp. 1047–1129**

**DOI: 10.4171/GGD/335**

Published online: 2015-11-16

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

Laurent Saloff-Coste^{[1]}and Tianyi Zheng

^{[2]}(1) Cornell University, Ithaca, United States

(2) Stanford University, USA

We study the decay of convolution powers of a large family $\mu_{S,a}$ of measures on finitely generated nilpotent groups. Here, $S=(s_1,\dots,s_k)$ is a generating $k$-tuple of group elements and $a=(\alpha_1,\dots,\alpha_k)$ 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 $s_i^{\pm m}$, $i=1,\dots,k,$ $m\in \mathbb N$. 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 $G_{j}=[G_{j-1},G]$, $G_1=G$.

*Keywords: *Random walk, stable laws, nilpotent groups

Saloff-Coste Laurent, Zheng Tianyi: Random walks on nilpotent groups driven by measures supported on powers of generators. *Groups Geom. Dyn.* 9 (2015), 1047-1129. doi: 10.4171/GGD/335