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

2020/03/17 by Thomas Doehrman, Doehrman, Thomas, Sunder Sethuraman +3
Mathematics · #60F05 #60G50 #FOS: Mathematics #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2003.07960

openalex publication_date 2020/03/17 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28

Abstract

We consider the scaling behavior of the range and p-multiple range, that is\nthe number of points visited and the number of points visited exactly p\≥ 1\ntimes, of simple random walk on mathbb Z^d, for dimensions d\≥ 2, up\nto time of exit from a domain D_N of the form D_N = ND where D\⊂\n mathbb R^d, as N uparrow\∞. Recent papers have discussed connections\nof the range and related statistics with the Gaussian free field, identifying\nin particular that the distributional scaling limit for the range, in the case\nD is a cube in d\≥ 3, is proportional to the exit time of Brownian\nmotion. The purpose of this note is to give a concise, different argument that\nthe scaled range and multiple range, in a general setting in d\≥ 2, both\nweakly converge to proportional exit times of Brownian motion from D, and\nthat the corresponding limit moments are `polyharmonic', solving a hierarchy of\nPoisson equations.\n

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