Self-intersections of Two-Dimensional Equilateral Random Walks and Polygons

TL;DR

This paper analyzes the mean and variance of self-intersections in two-dimensional equilateral random walks and polygons, revealing their expected counts grow as n log n and showing concentration around the mean.

Contribution

It provides the first detailed asymptotic analysis of self-intersections for equilateral random walks and polygons, including variance estimates.

Findings

Expected self-intersections grow as (2/π^2)n log n

Variance is O(n^2 log n), indicating concentration

Results apply to both walks and polygons with n steps

Abstract

We study the mean and variance of the number of self-intersections of the equilateral isotropic random walk in the plane, as well as the corresponding quantities for isotropic equilateral random polygons (random walks conditioned to return to their starting point after a given number of steps). The expected number of self-intersections is $(2/\pi^2)n\log n + O(n)$ for both walks and polygons with $n$ steps. The variance is $O(n^2 \log n)$ for both walks and polygons, which shows that the number of self-intersections exhibits concentration around the mean.