COMMUNICATIONS in PROBABILITY HEAT KERNEL ASYMPTOTICS ON THE LAMPLIGHTER GROUP

We show that, for one generating set, the on-diagonal decay of the heat kernel on the lamp-lighter group is asymptotic to c1n 1=6 exp[¡c2n1=3]. We also make o®-diagonal estimates which show that there is a sharp threshold for which elements have transition probabilities that are comparable to the return probability. The o®-diagonal estimates also give an upper bound for the heat kernel that is uniformly summable in time. The methods used also apply to a one dimensional trapping problem, and we compute the distribution of the walk conditioned on survival as well as a corrected asymptotic for the survival probability. Conditioned on survival, the position of the walker is shown to be concentrated within ®n1=3 of the origin for a suitable