Vicious Lévy flights

Abstract

We study the statistics of encounters of Lévy flights by introducing the concept of vicious Lévy flights - distinct groups of walkers performing independent Lévy flights with the process terminating upon the first encounter between walkers of different groups. We show that the probability that the process survives up to time $t$ decays as $t^{-α}$ at late times. We compute $α$ up to the second order in $ε$-expansion, where $ε=σ-d$, $σ$ is the Lévy exponent and $d$ is the spatial dimension. For $d=σ$, we find the exponent of the logarithmic decay exactly. Theoretical values of the exponents are confirmed by numerical simulations.