Persistence of some additive functionals of Sinai's walk

TL;DR

This paper investigates the probability that the sum of a function along Sinai's walk remains positive over time, revealing a specific decay rate linked to a persistence exponent originally identified in physics literature.

Contribution

It establishes the asymptotic probability decay for the positivity of additive functionals of Sinai's walk, connecting rigorous proofs with a physics-motivated persistence exponent.

Findings

Probability decays as 1/(log N)^{(3−√5)/2} for large N

Persistence exponent matches a value from physics literature

Results apply to a broad class of functions including f(x)=x

Abstract

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.