Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess