Wednesday, June 5, 2013

1306.0476 (Clélia de Mulatier et al.)

Asymmetric Lévy flights in the presence of absorbing boundaries    [PDF]

Clélia de Mulatier, Alberto Rosso, Gregory Schehr
We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution \phi(\eta) displaying asymmetric power law tails (i.e. \phi(\eta) \sim c/\eta^{\alpha +1} for large positive jumps and \phi(\eta) \sim c/(\gamma |\eta|^{\alpha +1}) for large negative jumps, with 0 < \alpha < 2). In absence of boundaries and after a large number of steps n, the probability density function (PDF) of the walker position, x_n, converges to an asymmetric L\'evy stable law of stability index \alpha and skewness parameter \beta=(\gamma-1)/(\gamma+1). In particular the right tail of this PDF decays as c n/x_n^{1+\alpha}. Much less is known when the walker is confined, or partially confined, in a region of the space. In this paper we first study the case of a walker constrained to move on the positive semi-axis and absorbed once it changes sign. In particular we compute exactly the persistence exponent \theta_+ which characterizes the algebraic large time decay of the survival probability and show that in this case the tail of the PDF of the walker position decays as c n/[(1-\theta_+) x_n^{1+\alpha}]. This last result can be generalized in higher dimensions such as a planar L\'evy walker confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations.
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