Convex Hulls of Multiple Random Walks: A Large-Deviation Study

TL;DR

This study investigates the large-deviation properties of convex hulls formed by multiple two-dimensional Gaussian random walks, revealing universal scaling laws and rate functions that follow the large-deviation principle.

Contribution

It introduces a large-deviation approach to analyze convex hulls of multiple random walks, providing new insights into their probability densities and scaling behaviors.

Findings

Densities exhibit universal scaling as functions of scaled area and perimeter.

Densities follow Gaussian distributions for perimeter and square root of area.

Rate functions obey a power law as the number of walks increases.

Abstract

We study the polygons governing the convex hull of a point set created by the steps of $n$ independent two-dimensional random walkers. Each such walk consists of $T$ discrete time steps, where $x$ and $y$ increments are i.i.d. Gaussian. We analyze area $A$ and perimeter $L$ of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below $10^{-900}$. We find that the densities exhibit a universal scaling behavior as a function of $A/T$ and $L/\sqrt{T}$, respectively. As in the case of one walker ($n=1$), the densities follow Gaussian distributions for $L$ and $\sqrt{A}$, respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for $T \rightarrow \infty$, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for $n \rightarrow \infty$ as found in the $n=1$ case. We also investigated the behavior of the averages as a function of the number of walks $n$ and found good agreement with the predicted behavior.