Exponents for the number of pairs of ${\alpha}$-favorite points of simple random walk in ${\mathbb Z}^2$

TL;DR

This paper explores the growth behavior of favorite points in a 2D simple random walk, revealing that their exponents match those of late and high points in the Gaussian free field, indicating deep connections between these phenomena.

Contribution

It establishes the asymptotic growth exponents for pairs of favorite points in 2D random walks, confirming conjectured relationships with Gaussian free field points.

Findings

Growth exponents of favorite points match those of late points.

Confirmed conjectured relationships between random walk favorite points and Gaussian free field.

Provided rigorous analysis of the asymptotic behavior of favorite points.

Abstract

We investigate a problem suggested by Dembo, Peres, Rosen, and Zeitouni, which states that the growth exponent of favorite points associated with a simple random walk in ${\mathbb Z}^2$ coincides, on average and almost surely, with those of late points and high points associated with the discrete Gaussian free field.