Rate of Decay of Concentration Functions on Discrete Groups

Given an irreducible probability measure on a non-compact locally compact group G, it is known that the concentration functions associated with converge to zero. In this note the rate of this convergence is presented in the case where G is a non-locally finite discrete group. In particular it is shown that if the volume growth V(m) of G satisfies V(m) cm ^D then for any compact set K we have sup _gG ⁽ⁿ⁾(Kg) Cn ^–D/2.