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Résumé |
The run-and-tumble particle, or persistent random walk, is one of the simplest non-Markovian random walk
model, which is currently of much interest, in particular in the context of active matter. In this talk, I will
consider an active run-and-tumble particle (RTP) in d dimensions and present exact results for the probability
S(t) that the x-component of the position of the RTP does not change sign up to time t. Remarkably, when the
tumblings occur at a constant rate, S(t) turns out to be independent of d for any finite time t (and not just for
large t), which is a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks
in one dimension. Moreover, 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. |
