Persistence of some additive functionals of Sinai's walk

2014/02/10 by Alexis Devulder, Devulder, Alexis
Mathematics · Physics and Astronomy · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR

paper · pdf · doi:10.48550/arxiv.1402.2267

30 pages, 2 figures

openalex publication_date 2014/02/10 · arxiv created 2015/03/09 · arxiv updated 2015/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are interested in Sinai's walk (S_n)_n∈ℕ. We prove that the annealed probability that ∑_k=0ⁿ f(S_k) is strictly positive for all n∈[1,N] is equal to 1/(log N)^{(3-√(5))/(2)+o(1)}, for a large class of functions f, and in particular for f(x)=x. The persistence exponent (3-√(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.

Citations

• Localization and number of visited valleys for a transient diffusion in random environment

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