Asymptotically exact probability distribution for the Sinai model with finite drift

TL;DR

This paper derives the exact asymptotic probability distribution for a biased Sinai model, revealing a creeping behavior of displacement moments over time, using a quantum diffusion mapping approach.

Contribution

It provides the first exact asymptotic distribution for the Sinai model with finite drift, employing a novel quantum diffusion mapping technique.

Findings

Displacement moments grow as t^{ b7} with time, indicating creeping behavior.

The long-time behavior is dominated by a lowest eigenvalue in an associated Schrf6dinger equation.

The method bridges stochastic processes and quantum diffusion for analyzing disordered systems.

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^{\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.