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 ⊂ R^d, as N ↑ ∞. 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.