Random walk in the low disorder ballistic regime

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.