Vicious Lévy Flights

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 $t^{-\alpha }$ at late times. We compute $\alpha$ up to the second order in $\epsilon$ expansion, where $\epsilon =\sigma -d$, $\sigma$ is the Lévy 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.

Figure 1

$\ln (S)$ vs $\ln (t)$ for two identical VLF in $d=1$. $\sigma$ values from top to bottom are 1.1, 1.3, 1.5, 1.7, and 2.5, respectively. Symbols represent simulation data and solid lines correspond to best fit lines to the late time data. Inset: Domains of VLF exponents in the $\sigma -d$ plane.

Figure 2

(a) 1-loop diagram corresponding to the integral $I_1$ (b), (c) 2-loop intgrals corresponding to $I_1^2$ and $I_2$ respectively.

Figure 3

$\alpha$ as a function of $\sigma$ for $d=1$. Symbols with error bars represent simulation data corresponding from top to bottom to $N=2$, 3, 4, VLF, respectively. Lines correspond to 1-loop approximation from formula (9) for $1<\sigma <2$. For $\sigma <1$, $\sigma \ge 2$ lines represent the mean field and Gaussian exponents, respectively. Inset: Same simulation data compared to 2-loop approximation from (9)

Figure 4

$\alpha$ vs $\sigma$ for predator and prey problem in $d=1$. Symbols represent simulation data for 4 predators and 1 prey (circles) and 3 predators and 2 prey (squares). Lines are 2-loop approximation from formula (9).