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