Universal Survival Probability for a d -Dimensional Run-and-Tumble Particle

Francesco Mori (1) , Pierre Le Doussal (2) , Satya Majumdar (1) , Gregory Schehr (1)

1 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
2 Champs Aléatoires et Systèmes hors d'Équilibre

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.

Domaines: Physique [physics]

Francesco Mori, Pierre Le Doussal, Satya Majumdar, Gregory Schehr. Universal Survival Probability for a d -Dimensional Run-and-Tumble Particle. Physical Review Letters, 2020, 124 (9), ⟨10.1103/PhysRevLett.124.090603⟩. ⟨hal-02512214⟩