CONVERGENCE IN LAW FOR THE CAPACITY OF THE RANGE OF A CRITICAL BRANCHING RANDOM WALK

Let R-n be the range of a critical branching random walk with n particles on Z(d) , which is the set of sites visited by a random walk indexed by a critical Galton- Watson tree conditioned on having exactly n vertices. For d is an element of {3, 4, 5}, we prove that n - (d-2)/(4) cap((d)) (R-n), the renormalized capacity of Rn, converges in law to the capacity of the support of the integrated super-Brownian excursion. The proof relies on a study of the intersection probabilities between the critical branching random walk and an independent simple random walk on Z( d) .