Random walks, word metric and orbits distribution on the plane

Given a countably infinite group $G$ acting on some space $X$, an increasing family of finite subsets $G_n$ and $x\in X$, a natural question to ask is what asymptotical distribution the sets $G_nx$ form. More formally, we define for a function $f$ over $X$ the sums $S_n(f,x)=\sum_{g\in G_n}f(gx)$ and ask whether exists a function $\Psi(n):\mathbb{N}\to\mathbb{R}$ such that the sequence $\Psi(n)S_n(f,x)$ converges. This is a delicate problem that was studied under various settings. We first show a full solution when elements are chosen using a carefully chosen word metric from a specific lattice in $SL(2,\mathbb{Z})$ acting on the circle. In addition, it is proven that the resulting measure is stationary with respect to a certain random walk and has a tight connection to a well studied function from the field of Diophantine approximations. We then proceed to study the asymptotic distribution problem when elements are chosen using a random walk over $SL(2,\mathbb{R})$ acting on $\mathbb{R}^2$. We offer a variant of our initial problem which yields some surprising and interesting results.