Exact Results on Sinai's Diffusion

TL;DR

This paper analyzes Sinai's diffusion problem in one dimension, using stochastic representations to confirm known results and demonstrate the dominance of disorder in the scaling regime, leading to zero-one laws.

Contribution

It introduces a stochastic representation method that reproduces known results and confirms recent findings in Sinai's diffusion problem.

Findings

Disorder dominates the diffusion in the Sinai scaling regime.

Thermal distributions tend to zero-one laws under strong disorder.

The method reproduces rigorous results and confirms recent approximation-based results.

Abstract

We study the continuum version of Sinai's problem of a random walker in a random force field in one dimension. A method of stochastic representations is used to represent various probability distributions in this problem (mean probability density function and first passage time distributions). This method reproduces already known rigorous results and also confirms directly some recent results derived using approximation schemes. We demonstrate clearly, in the Sinai scaling regime, that the disorder dominates the problem and that the thermal distributions tend to zero-one laws.