Remarks on the range and multiple range of a random walk up to the time of exit

Abstract

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 ℤd, for dimensions d≥2, up to time of exit from a domain DN of the form DN=ND, where D⊂ℝd, 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.