Constants of concentration for a simple recurrent random walk on random environment

TL;DR

This paper investigates the asymptotic behavior of Sinai's walk, providing improved results on the concentration neighborhood size and the frequency of visited points over time.

Contribution

It refines existing asymptotic results for Sinai's walk and establishes the almost sure limit of the number of frequently visited points.

Findings

Improved asymptotic bounds for the neighborhood of concentration.

Determined the almost sure limit of frequently visited points.

Enhanced understanding of Sinai's walk behavior.

Abstract

We clarify the asymptotic of the limsup of the size of the neighborhood of concentration of Sinai's walk improving the result in \cite{Pierre3}. Also we get the almost sure limit of the number of points visited more than a small but fixed proportion of a given amount of time.