Asymptotically exact probability distribution for the Sinai model with finite drift
arXiv:1007.5287 · doi:10.1103/PhysRevE.82.030103
Abstract
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, <x^n> ~ t^{μn} where μis 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.
arXiv comment4 pages, 2 figures
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