Random walks, word metric and orbits distribution on the plane

TL;DR

This paper investigates the asymptotic distribution of orbits generated by group actions on the plane, providing solutions for specific lattice cases and exploring random walk scenarios with connections to Diophantine approximation.

Contribution

It offers a complete solution for orbit distribution using word metrics on a lattice in SL(2,Z) and extends analysis to random walks on SL(2,R), revealing new distribution behaviors.

Findings

Full solution for lattice-based orbit distribution

Stationary measure related to Diophantine approximation

Surprising results in random walk orbit analysis

Abstract

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.