On large deviations for the range of a two-dimensional random walk

TL;DR

This paper analyzes the probability of a two-dimensional symmetric random walk visiting significantly more vertices than expected, focusing on deviations between average behavior and linear growth, providing insights into large deviation probabilities.

Contribution

It introduces precise calculations for large deviation probabilities of the range of a 2D random walk, filling a gap in understanding intermediate scale deviations.

Findings

Derived explicit large deviation estimates for the walk's range

Identified the scale of deviations between mean and linear growth

Enhanced understanding of the walk's range behavior in two dimensions

Abstract

In this note, we compute the probability that a two-dimensional symmetric random walk visits more vertices than expected, for deviations on scales between the mean behavior and linear growth.