Topics and problems on favorite sites of random walks

TL;DR

This paper investigates special points like favorite, late, and high points in simple random walks and Gaussian free fields, extending previous results, proposing open problems, and surveying their geometric structures in two dimensions.

Contribution

It extends existing results on favorite points and related phenomena in random walks, and provides a comprehensive survey of their geometric structures in two dimensions.

Findings

Extended results on favorite points in 2D random walks.

Proposed open problems for the case d=2.

Surveyed geometric structures of special points.

Abstract

In this article, we study special points of a simple random walk and a Gaussian free field, such as (nearly) favorite points, late points and high points. In section $2$, we extend results of [19] and suggest open problems for $d=2$. In section $3$, we give a survey on the geometric structures of (nearly) favorite points and late points of a simple random walk and high points of a Gaussian free field in two dimension.