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- W3011806287 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 $pgeq 1$ 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." @default.
- W3011806287 created "2020-03-23" @default.
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- W3011806287 date "2020-03-17" @default.
- W3011806287 modified "2023-09-27" @default.
- W3011806287 title "REMARKS ON THE RANGE AND MULTIPLE RANGE OF RANDOM WALK UP TO THE TIME OF EXIT" @default.
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