Rate of Decay of Concentration Functions on Discrete Groups

Given an irreducible probability measure μ on a non-compact locally compact group G, it is known that the concentration functions associated with μ 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 _gεGμ ⁽ⁿ⁾(Kg) ≤ Cn ^-D/2.