Jonathan. Frech’s WebBlog

Mostly Misaligned Mirrors (#215)

Jonathan Frech,

Recently, my stochastic professor introduced me to a prob­lem he has been pondering for over two decades: on the two-di­men­sion­al in­te­ger lattice $\mathbb{Z}^2$$\mathbb{Z}^2$$\mathbb{Z}^2$ one shall flip a three-sided coin for each point and uni­form­ly place one of three mirrors, $\{\diagup,\,\cdot\,,\diagdown\}$$\{\diagup,\,\cdot\,,\diagdown\}$$\{\diagup,\,\cdot\,,\diagdown\}$, where $\,\cdot\,$$\,\cdot\,$$\,\cdot\,$ denotes not placing a mirror. After having populated the world, one picks their favorite in­te­ger tuple and points a beam of light in one of the four cardinal directions. With what probability does the light fall into a loop, never fully escaping?

A cycle which in­cludes the origin.
-=-

With me almost surely not being able to solve this prob­lem thru proof, I took the route of sil­i­con-assisted prob­lem anal­y­sis: simulation. How­ev­er, even on very large lattice segments, the beam often ei­ther escapes or exists on­ly for a very short time — as seen above.
Source code: mostly-misaligned-mirrors_visualize.py, mostly-misaligned-mirrors_simulate.py

An example of a rather in­tri­cate path which eventually escapes the cho­sen view:

The beam escapes on a 1000 ⨉ 1000 board.

After seeing the beams often escape my view frames, I decided to also simulate the prob­lem with­out visualization — both to potentially get a per­for­mance increase as well as to lift the constraints of fi­nite imagery. My results for run­ning ten simulations are as follows:

$ python mostly-misaligned-mirrors_simulate.py 10
[X] Exceeded the maximum visited mirrors threshold of 1000000.
[X] Exceeded the maximum visited mirrors threshold of 1000000.
[.] Visited 5110 mirrors before falling into a loop.
[X] Exceeded the maximum visited mirrors threshold of 1000000.
[.] Visited 4 mirrors before falling into a loop.
[X] Exceeded the maximum visited mirrors threshold of 1000000.
[X] Exceeded the maximum visited mirrors threshold of 1000000.
[X] Exceeded the maximum visited mirrors threshold of 1000000.
[X] Exceeded the maximum visited mirrors threshold of 1000000.
[X] Exceeded the maximum visited mirrors threshold of 1000000.

One can clear­ly see that whilst some paths are of minuscule lengths, others far exceed those, traversing over a mil­lion mirrors until getting stopped as to not fry my ma­chine.

My professor’s conjecture states that any given mirror configuration almost surely — meaning with probability 1 — causes the beam of light to fall into a cycle. In some sense it appears intuitively clear — escaping re­quires an infinite amount of mirrors to be placed correctly, how­ev­er: could the beam also find a sneaky way around a lot (in the sense of being of non-vanishing measure) of configurations, squeezing elegantly and indefinitely around all obstacles? After run­ning my simulations doubts in the former increase, leaving me cluelessly in the dark, waiting for a proper proof to be formalized.