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 µS,a of measures on finitely generated nilpotent groups. Here, S = (s1, . . . , s k ) is a generating k-tuple of group elements and a = (α1, . . . , α k ) is a k-tuple of reals in the interval (0, 2). The symmetric measure µS,a is supported bym ∈ Z} and gives probability proportional to (1 + m) −α i −1 to s ±m i , i = 1, . . . , k, m ∈ N. We determine the behavior of the probability of return µ (n) S,a (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 si within the lower central series Gj = [Gj−1, G], G1 = G.

Random walks on finite nilpotent groups driven by long-jump measures

“…Organization This paper is an extension of work done in [12,1]; these papers are concerned with infinite groups, whereas this paper studies finite groups. Many of the techniques used in here take inspiration from proofs from those two papers; we will give specific citations as we use them.…”

“…In Section 2, we start by proving Theorem 1.3. We prove the upper bound by using a pseudo-Poincaré inequality, where we rely heavily on results developed in [12]. For the lower bound, we use the Courant-Fischer characterization of β 1 with a test function and bounds that are similar to those in [1].…”

“…For the lower bound, we use the Courant-Fischer characterization of β 1 with a test function and bounds that are similar to those in [1]. In Section 3, we prove the doubling property, i.e., Theorem 1.6 using growth results from [12] for free nilpotent groups and a lemma of [6] to transport the result to finite nilpotent groups.…”

“…Theorem 2.1. [12,Theorem 2.10] Let k and be positive integers and a ∈ (0, 2) k . There exist C = C( , k, a), p = p( , k, a), and (i 1 , .…”

“…Proof. Let Ĝ = N ( , k) be the free nilpotent group of nilpotency class and generated by S. Theorem 2.10 of [12] states that there exist an integer p = p( , k, a), a constant C = C( , k, a), and (i 1 , . .…”