Vicious L\'evy flights

TL;DR

This paper investigates the encounter statistics of vicious Le9vy flights, revealing decay behaviors and exponents through analytical calculations and numerical simulations, advancing understanding of complex stochastic processes.

Contribution

It introduces the concept of vicious Le9vy flights and computes the decay exponents of survival probability up to second order in b5-expansion, including exact results at critical dimension.

Findings

Survival probability decays as t^{-b1} at late times.

Analytical exponents match numerical simulation results.

Exact logarithmic decay exponent found at critical dimension d=c3.

Abstract

We study the statistics of encounters of L\'evy flights by introducing the concept of vicious L\'evy flights - distinct groups of walkers performing independent L\'evy 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 L\'evy 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.