ANNEALED LOCAL LIMIT THEOREM FOR SINAI'S RANDOM WALK IN RANDOM ENVIRONMENT

Alexis Devulder (1)

1 LMV - Laboratoire de Mathématiques de Versailles

We consider Sinai's random walk in random environment (S_n)_{n∈N}. We prove a local limit theorem for (S_n)_{n∈N} under the annealed law P. As a consequence, we get an equivalent for the annealed probability 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 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.

Mots clés: Sinai's walk; random walk; random environment; local limit theorem; localization; path decomposition; random walks conditioned to stay positive; renewal theory

Domaines: Probabilités [math.PR]

Alexis Devulder. ANNEALED LOCAL LIMIT THEOREM FOR SINAI'S RANDOM WALK IN RANDOM ENVIRONMENT. 2022. ⟨hal-03851879v1⟩