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

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 \(G_{j}=[G_{j-1},G]\), \(G_1=G\).