TITLE: Random reflections unexpectedly produce banded distributions
QUESTION [8 upvotes]: Start with $p_1$ a random point on the origin-centered unit circle $C$.
At step $i$, select a random point $q_i$ on $C$, and a random mirror line
$M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to point $p_{i+1}$.
(Note that if every $M_i$ passed through the origin rather than $q_i \in C$,
all $p_i$ would lie on $C$.)
In the example below, $p_1$ on $C$ reflects to $p_2$ near the origin,
$p_2$ reflects out to $p_3$ near $(1,1.5)$, etc.

          


          

Green connects $p_1,\ldots,p_5$. Dark lines are mirrors $M_i$.


I expected that repeating this process would produce some smooth
distribution around the origin, as the points reflect further and further out,
when the mirrors effectively pass through the origin at that expanded scale.
But I find not infrequently bands of density, as depicted below.

   


   

Each distribution includes $n=5000$ reflected points. The unit circle is red.
Approx. radii: $107,67,47,148$.


Extending $n$ to larger values seems to continue banding effects.
Here is the 3rd example above, the most uniform of those four, extended.
Its radius increases from $47$ to $225$:

          


          

Continuation of 3rd example above, to $n=25000$. Approx. radius $225$.



Q. What explains this radial clustering/banding behavior?

REPLY [2 votes]: As said by Sangchui Lee, we are interested in $r_i = |p_i|$ and we assume that the reflections induce enougth random rotation such that the "bands" of radius $r$ just reveal the number of time $r_i$ was closed to $r$. And we have
$$| p_{i+1}| =|p_i -2\langle q_i, n_i \rangle n_i|$$ 
We define then a random march $X_i$ defined by $X_0=0$ and $$X_{i+1}=X_i-2\langle q_i, n_i \rangle n_i$$
We immediately have that $|p_i|$ has the same law as $|X_i|$ and that for large $i$, $X_i$ behave like the brownian motion on $\mathbb{R}^2$.
One can calculate that the variance of $2\langle q_i, n_i \rangle n_i$ is $\sigma^2 = 2$, the points should be then in a disk of randius $\sim \sqrt{n \sigma^2 }=100$ for $n=5000$ and $224$ for $n=25000$ . 
Conclusion for large $i$, $|p_i|$ is given by the Bessel process. And as Sangchui Lee said we are interested in the local time. 
https://en.wikipedia.org/wiki/Local_time_(mathematics). 
The Ray-Knigh theorem should be usefull here even if it is state for the Brownian motion and not for the Bessel process.
(original paper of Knigh https://www.jstor.org/stable/pdf/1993647.pdf)