Persistence of some additive functionals of Sinai's walk

Alexis Devulder (1)

1 LMV - Laboratoire de Mathématiques de Versailles

We are interested in Sinai's walk $(S_n)_{n\in\mathbb{N}}$. We prove that the annealed probability that $\sum_{k=0}^n f(S_k)$ is strictly positive for all $n\in[1,N]$ is equal to $1/(\log N)^{\frac{3-\sqrt{5}}{2}+o(1)}$, for a large class of functions $f$, and in particular for $f(x)=x$. The persistence exponent $\frac{3-\sqrt{5}}{2}$ first appears in a non-rigorous paper of Le Doussal, Monthus and Fischer, with motivations coming from physics. The proof relies on techniques of localization for Sinai's walk and uses results of Cheliotis about the sign changes of the bottom of valleys of a two-sided Brownian motion.

Mots clés: persistence; survival exponent; one-sided exit problem; Random walk in random environment; Sinai's walk; integrated random walk

Domaines: Probabilités [math.PR]

Alexis Devulder. Persistence of some additive functionals of Sinai's walk. 2014. ⟨hal-00944631v2⟩