Mixing of a generic simple symmetric random walk on the circle

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.