Remarks on the range and multiple range of 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 $pgeq 1$ times, of simple random walk on ${mathbb Z}^d$, for dimensions $dgeq 2$, up to time of exit from a domain $D_N$ of the form $D_N = ND$ where $Dsubset {mathbb R}^d$, as $Nuparrowinfty$. 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 $dgeq 3$, 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 $dgeq 2$, 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.