Vicious Levy Flights

We study the statistics of encounters of Levy flights by introducing the concept of vicious Levy flights-distinct groups of walkers performing independent Levy 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(-alpha) at late times. We compute alpha up to the second order in epsilon expansion, where epsilon = sigma - d, sigma is the Levy exponent, and d is the spatial dimension. For d = sigma, we find the exponent of the logarithmic decay exactly. Theoretical values of the exponents are confirmed by numerical simulations. Our results indicate that walkers with smaller values of sigma survive longer and are therefore more effective at avoiding each other.