Universal survival probability for a $d$-dimensional run-and-tumble particle

Abstract

We consider an active run-and-tumble particle (RTP) in $d$ dimensions and compute exactly the probability $S(t)$ that the $x$-component of the position of the RTP does not change sign up to time $t$. When the tumblings occur at a constant rate, we show that $S(t)$ is independent of $d$ for any finite time $t$ (and not just for large $t$), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed $v$ of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem.

Main text: 5 pages + 2 figs., Supp. Mat: 12 pages + 4 figs

References in corpus (12)

• Novel type of phase transition in a system of self-driven particles
• Motility-Induced Phase Separation
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics?
• Active matter
• Universal Record Statistics of Random Walks and Lévy Flights
• First-passage time of run-and-tumble particles
• First-passage and first-hitting times of Levy flights and Levy walks
• Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap
• Record statistics and persistence for a random walk with a drift
• Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results
• Non-crossing run-and-tumble particles on a line

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