Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study

TL;DR

This paper investigates the large deviation properties of the volume and surface area of convex hulls formed by random walks in higher dimensions, using numerical methods to explore rare events and scaling behaviors.

Contribution

It introduces a numerical approach to analyze large deviations in convex hull properties of random walks in dimensions three and four, extending understanding beyond mean behaviors.

Findings

Distribution scaling with walk length T similar to 2D case

Behavior of distribution tails characterized

Means confirmed and variances calculated for large T

Abstract

The distribution of the hypervolume $V$ and surface $\partial V$ of convex hulls of (multiple) random walks in higher dimensions are determined numerically, especially containing probabilities far smaller than $P = 10^{-1000}$ to estimate large deviation properties. For arbitrary dimensions and large walk lengths $T$, we suggest a scaling behavior of the distribution with the length of the walk $T$ similar to the two-dimensional case, and behavior of the distributions in the tails. We underpin both with numerical data in $d=3$ and $d=4$ dimensions. Further, we confirm the analytically known means of those distributions and calculate their variances for large $T$.