  Fix an irrational number $\alpha$ and a real function $\mathfrak{p}$ on the
circle with $0<\mathfrak{p}<1$. If a particle is placed at a point $x\in
\mathbb R/\mathbb Z$, then in the next step it jumps to $x+\alpha$ with
probability $\mathfrak{p}(x)$ and to $x-\alpha$ with probability
$1-\mathfrak{p}(x)$. Sinai and Kaloshin proved that if $\mathfrak{p}$ is smooth
then the random walk is uniquely ergodic and mixing, unless $\alpha$ is
Liouville and $\mathfrak{p}$ is symmetric. Unique ergodicity in the general
case has been obtained by Conze and Guivarc'h. Here we give an alternative
proof of the latter as well as some generic result about mixing, which
partially solves a recent open problem.
