Random walk in the low disorder ballistic regime

Abstract

We consider a random walk in ${\mathbb{Z}}^d$ which jumps from a site $x$ to a nearest neighboring site $x+e$ (where $e\in V:=\{x\in {\mathbb{Z}}^d:|x{|}_1=1\}$) with probability $p_0(e)+\epsilon \xi (x,e)$. Here ${\sum }_e{p}_0(e)=1$, $p_0(e)>0$, $\epsilon$ is a small parameter while $\{\{\xi (x,e):e\in V\}:x\in {\mathbb{Z}}^d\}$ are i.i.d. random variables with an absolute value bounded by $1$. We review recent progress in the non-vanishing velocity case, giving an asymptotic expansion in $\epsilon$ of the invariant measure of the environmental process, and bounds for the velocity.

References in corpus (2)

• Ballistic random walks in random environment at low disorder
• Effective Polynomial Ballisticity Condition for Random Walk in Random Environment