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 $p\ge 1$ times, of a simple random walk on ${\mathbb{Z}}^d$, for dimensions $d\ge 2$, up to time of exit from a domain $D_N$ of the form $D_N=ND$, where $D\subset {ℝ}^d$, as $N\uparrow \infty$. 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 $d\ge 3$, 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 $d\ge 2$, 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.