JournalsggdVol. 9, No. 4pp. 1047–1129

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

  • Laurent Saloff-Coste

    Cornell University, Ithaca, United States
  • Tianyi Zheng

    Stanford University, USA
Random walks on nilpotent groups driven by measures supported on powers of generators cover
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Abstract

We study the decay of convolution powers of a large family μS,a\mu_{S,a} of measures on finitely generated nilpotent groups. Here, S=(s1,,sk)S=(s_1,\dots,s_k) is a generating kk-tuple of group elements and a=(α1,,αk)a=(\alpha_1,\dots,\alpha_k) is a kk-tuple of reals in the interval (0,2)(0,2). The symmetric measure μS,a\mu_{S,a} is supported by S={sim,1ik,mZ}S^*=\{s_i^{m}, 1\le i\le k,\,m\in \mathbb Z\} and gives probability proportional to (1+m)αi1(1+m)^{-\alpha_i-1} to si±ms_i^{\pm m}, i=1,,k,i=1,\dots,k, mNm\in \mathbb N. We determine the behavior of the probability of return μS,a(n)(e)\mu_{S,a}^{(n)}(e) as nn tends to infinity. This behavior depends in somewhat subtle ways on interactions between the kk-tuple aa and the positions of the generators sis_i within the lower central series Gj=[Gj1,G]G_{j}=[G_{j-1},G], G1=GG_1=G.

Cite this article

Laurent Saloff-Coste, Tianyi Zheng, Random walks on nilpotent groups driven by measures supported on powers of generators. Groups Geom. Dyn. 9 (2015), no. 4, pp. 1047–1129

DOI 10.4171/GGD/335