The convex hull of the run-and-tumble particle in a plane – Archive ouverte HAL

Alexander K HartmannSatya N Majumdar 1 Hendrik Schawe 2 Gregory Schehr 1 Alexander Hartmann 2 Satya Majumdar 1

Alexander K Hartmann, Satya N Majumdar, Hendrik Schawe, Gregory Schehr, Alexander Hartmann, et al.. The convex hull of the run-and-tumble particle in a plane. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2020, 2020 (5), pp.053401. ⟨10.1088/1742-5468/ab7c5f⟩. ⟨hal-02881103⟩

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.

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. University of Oldenburg