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

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 $z_n=O((\log n{)}^2)$. 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.