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

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.

arXiv comment21 pages, 8 Figures, 1 Table

References in corpus (29)

• Novel type of phase transition in a system of self-driven particles
• Active Particles in Complex and Crowded Environments
• The large deviation approach to statistical mechanics
• Motility-Induced Phase Separation
• Active Brownian Particles. From Individual to Collective Stochastic Dynamics
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics?
• Active matter
• Flocking at a distance in active granular matter
• Run-and-tumble particle in one-dimensional confining potential: Steady state, relaxation and first passage properties
• Random Convex Hulls and Extreme Value Statistics
• Jamming and Attraction of Interacting Run-and-Tumble Random Walkers
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• Spatial extent of an outbreak in animal epidemics
• Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap
• Stationary superstatistics distributions of trapped run-and-tumble particles
• Exact spectral solution of two interacting run-and-tumble particles on a ring lattice
• Convex Hulls of Random Walks: Large-Deviation Properties
• Volumetric Properties of the Convex Hull of an n-dimensional Brownian Motion
• The convex hull for a random acceleration process in two dimensions
• Mean perimeter and mean area of the convex hull over planar random walks
• Survival probability of a Brownian motion in a planar wedge of arbitrary angle
• Ground state energy of noninteracting fermions with a random energy spectrum
• Large Deviations of Convex Hulls of Self-Avoiding Random Walks
• Rare-Event Properties of the Nagel-Schreckenberg Model
• Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study
• Mean perimeter of the convex hull of a random walk in a semi-infinite medium
• Large Deviations of Convex Hulls of the "True" Self-Avoiding Random Walk
• Score distributions of gapped multiple sequence alignments down to the low-probability tail

Cited by in corpus (31)

• Condensation transition in the late-time position of a Run-and-Tumble particle
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Position distribution in a generalised run and tumble process
• Survival probability of a run-and-tumble particle in the presence of a drift
• AOUP in the presence of Brownian noise: a perturbative approach
• Mean area of the convex hull of a run and tumble particle in two dimensions
• Time to reach the maximum for a stationary stochastic process
• Run-and-Tumble particle in inhomogeneous media in one dimension
• Thermodynamic work of partial resetting
• Heat fluctuations in a harmonic chain of active particles
• Local time for run and tumble particle
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• Attractor-driven matter
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• Generating constrained run-and-tumble trajectories
• Jamming of multiple persistent random walkers in arbitrary spatial dimension
• Confined run and tumble particles with non-Markovian tumbling statistics
• Tuning attraction and repulsion between active particles through persistence
• From a microscopic solution to a continuum description of active particles with a recoil interaction in one dimension
• Extremal statistics for a one-dimensional Brownian motion with a reflective boundary
• Generalized arcsine laws for a sluggish random walker with subdiffusive growth
• Large deviations in statistics of the convex hull of passive and active particles: A theoretical study
• A unified perspective on exponential tilt and bridge algorithms for rare trajectories of discrete Markov processes
• Exact joint distributions of three global characteristic times for Brownian motion
• Optimal finite-differences discretization for the diffusion equation from the perspective of large-deviation theory
• Tracer dynamics in the active random average process
• Long time behavior of run-and-tumble particles in two dimensions
• Run-and-tumble particles on a line with a fertile site
• Crystal to liquid cross-over for active particles with inverse-square power-law interaction
• Universal framework for the long-time position distribution of free active particles
• Sampling rare trajectories using stochastic bridges