Annealed local limit theorem for Sinai's random walk in random environment

TL;DR

This paper establishes a local limit theorem for Sinai's one-dimensional random walk in a random environment under the annealed measure, providing precise asymptotics for the walk's position when scaled logarithmically.

Contribution

It introduces a new path decomposition technique for Sinai's potential and derives an annealed local limit theorem with detailed environment and trajectory analysis.

Findings

Proves an annealed local limit theorem for Sinai's walk.

Provides asymptotic estimates for probabilities of the walk's position.

Develops a novel path decomposition for the potential of Sinai's walk.

Abstract

We consider Sinai's random walk in random environment $(S_n)_{n\in\mathbb{N}}$. We prove a local limit theorem for $(S_n)_{n\in\mathbb{N}}$ under the annealed law $\mathbb{P}$. As a consequence, we get an equivalent for the annealed probability $\mathbb{P}(S_n=z_n)$ as $n$ goes to infinity, when $z_n=O\big((\log n)^2\big)$. To this aim, we develop a path decomposition for the potential of Sinai's walk, that is, for some random walks with i.i.d. increments. The proof also relies on renewal theory, a coupling argument, a very careful analysis of the environments and trajectories of Sinai's walk satisfying $S_n=z_n$, and on precise estimates for random walks conditioned to stay positive or nonnegative.