Asymptotically exact probability distribution for the Sinai model with finite drift

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, $⟨x^n⟩\sim t^{\mu n}$, where $\mu <1$ 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 Schrödinger equation. For nonzero drift such an equation has an isolated lowest eigenvalue separated by a gap from quasicontinuous excited states, and the eigenstate corresponding to the former governs the long-time asymptotic behavior.

Figure 1

The stochastic variable ${\xi }^{-}\left(x^{\prime }\right)$ depends on realization of random drifts for the “earlier time” $\left(<x\right)$ while ${\xi }^{+}\left(x\right)$ for the “later” $\left(>x\right)$. Therefore, blocks built of ${\xi }^{+}$ and ${\xi }^{-}$ are statistically independent.

Figure 2

The effective potential for the Schrödinger equation.