The convex hull of the run-and-tumble particle in a plane

TL;DR

This paper analyzes the statistical properties of the convex hull of a planar run-and-tumble particle, deriving exact average perimeter expressions and exploring the full distribution, including rare events, using numerical methods.

Contribution

It provides exact formulas for the average perimeter of the convex hull of a run-and-tumble particle and characterizes the full distribution, including rare tail events, in different statistical ensembles.

Findings

Exact average perimeter expressions derived

Full distribution computed with high precision

Rare event analysis of the convex hull tails

Abstract

We study the statistical properties of the convex hull of a planar run-and-tumble particle (RTP), also known as the "persistent random walk", where the particle/walker runs ballistically between tumble events at which it changes its direction randomly. We consider two different statistical ensembles where we either fix (i) the total number of tumblings $n$ or (ii) the total duration $t$ of the time interval. In both cases, we derive exact expressions for the average perimeter of the convex hull and then compare to numerical estimates finding excellent agreement. Further, we numerically compute the full distribution of the perimeter using Markov chain Monte Carlo techniques, in both ensembles, probing the far tails of the distribution, up to a precision smaller than $10^{-100}$. This also allows us to characterize the rare events that contribute to the tails of these distributions.