REMARKS ON THE RANGE AND MULTIPLE RANGE OF A RANDOM WALK UP TO THE TIME OF EXIT

We consider the scaling behavior of the range and p-multiple range, that is the number of points visited and the number of points visited exactly p >= 1 times, of a simple random walk on Z(d), for dimensions d >= 2, up to time of exit from a domain D-N of the form D-N = ND, where D subset of R-d, as N up arrow infinity. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case D is a cube in d >= 3, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in d >= 2, both weakly converge to proportional exit times of Brownian motion from D, and that the corresponding limit moments are "polyharmonic", solving a hierarchy of Poisson equations.