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

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). symmetric measure mu(S,a) is supported by S* = {s(i)(m), 1 <= i <= k, m is an element of Z} and gives probability proportional to (1 + m)(-alpha i-1) to s(i)(+/- m), i = 1, . . ., k, m is an element of N. We determine the behavior of the probability of return mu((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 s(i) within the lower central series G(j) = [G(j-1), G], G(1) = G.