A Local Limit Theorem and Loss of Rotational Symmetry of Planar Symmetric Simple Random Walk

TL;DR

This paper establishes a refined local limit theorem for symmetric simple random walks on 1D and 2D lattices, revealing a transition from rotational symmetry to axis concentration as deviations grow.

Contribution

It provides explicit asymptotic formulas for the probability distribution of the walk's position, improving previous results for distant points and describing symmetry loss in planar walks.

Findings

Asymptotic expressions for $P(S_n=x)$ valid for all $x$.

Planar symmetric random walk loses rotational symmetry outside radius $n^{3/4}$.

Walks tend to align along axes when far from the origin.

Abstract

We derive a local limit theorem for normal, moderate, and large deviations for symmetric simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the walk's starting point. More specifically, we give explicit asymptotic expressions in terms of $n$ and $x$, where $x$ is thought of as dependent on $n$, in dimensions one and two for $P(S_n=x)$, the probability that symmetric simple random walk $S$ started at the origin is at some point $x$ at time $n$, that are valid for all $x$. We also show that the behavior of planar symmetric simple random walk differs radically from that of planar standard Brownian motion outside of the disk of radius $n^{3/4}$, where the random walk ceases to be approximately rotationally symmetric. Indeed, if $n^{3/4}=o(|S_n|)$, $S_n$ is more likely to be found along the coordinate axes. In this paper, we give a description of how the transition from approximate rotational symmetry to complete concentration of $S$ along the coordinate axes occurs.