Limit Theorem for sub-ballistic random walks in Dirichlet environment in dimension $d\geq 3$

TL;DR

This paper proves that sub-ballistic random walks in Dirichlet environments in dimensions $d extgreater=3$ converge, after renormalization, to a $ ext{ extkappa}$-stable asymmetric Levy process, revealing detailed asymptotic behavior.

Contribution

It establishes the convergence of sub-ballistic Dirichlet environment random walks to a stable Levy process, extending understanding of their long-term behavior.

Findings

Walks converge to a $ ext{ extkappa}$-stable Levy process after renormalization

Displacement is polynomial of order $ ext{ extkappa}$ in sub-ballistic regime

Provides explicit conditions for convergence in high dimensions

Abstract

We look at random walks in Dirichlet environment. It was known that in dimension $d\geq 3$, if the walk is sub-ballistic, the displacement of the walk is polynomial of order $\kappa$ for some explicit $\kappa$. We show that the walk, after renormalization, actually converges to a $\kappa$-stable completely asymmetric Levy Process.