Geometric structures of late points of a two-dimensional simple random walk

TL;DR

This paper investigates the geometric structure of late points in a two-dimensional simple random walk, comparing their properties with nearly favorite points and high points in the Gaussian free field, and provides new estimates for multipoint sets.

Contribution

It offers novel estimates for the exponents of multipoint late points, highlighting differences from related point classes and advancing understanding of their geometric structures.

Findings

Exponents for multipoint late points are estimated in average.

Differences are identified between late points and other point classes.

Results reveal unique geometric structures of late points.

Abstract

We consider the problem, as suggested by Dembo ($2003$, $2006$), of late points of a simple random walk in two dimensions. It has been shown that the exponents for the numbers of pairs of late points coincide with those of nearly favorite points and high points in the Gaussian free field, whose exact values are known. We estimate the exponents for the numbers of a multipoint set of late points in average. While there have been observed certain similarities between among three classes of points, our results exhibit a difference.