Asymptotically exact probability distribution for the Sinai model with finite drift

Details Asymptotically exact probability distribution for the Sinai model with finite drift Asymptotically exact probability distribution for the Sinai model with finite drift

2010-07-29

arxiv.org/abs/1007.5287v1

Phys. Rev. E 82, 030103(R) (2010)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 2 figures

Scientific paper

10.1103/PhysRevE.82.030103

We obtain the exact asymptotic result for the disorder-averaged probability distribution function for a random walk in a biased Sinai model and show that it is characterized by a creeping behavior of the displacement moments with time, ~ t^{\mu n} where \mu is dimensionless mean drift. We employ a method originated in quantum diffusion which is based on the exact mapping of the problem to an imaginary-time Schr\"{odinger} equation. For nonzero drift such an equation has an isolated lowest eigenvalue separated by a gap from quasi-continuous excited states, and the eigenstate corresponding to the former governs the long-time asymptotic behavior.

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