# Is there an unbiased random walk on a colored plane for any number of colors?

So I was trying to motivate the fundamental postulate of statistical mechanics (i.e. all microstates are assumed to be equally probable $-$ even if we can't practically measure them, but only their macroscopic properties) using simple examples.

I came up with the simple analog of a dice, where the six sides $1-6$ are the random microstates and a yes/no-answer machine can only compute some rough "macroscopic" property of the result.

When I tried to incorporate a metaphor for the dynamics, which changes one microstate to another, a mathematical problem emerged:

The idea is to create a two-dimensional plane,

$\ \ \$ (example of a random plane)

with $n$ (e.g. six) differently colored fields and the following property: If you stand on one field of some color and you take a random step to a neightboring field you can end up on any of the other colors. So the problem is to find a colored plane, finite or not, such that every other color can be reached from every colored field. The number of colors $n$ is fixed, but the number of fields on the plane is $n$ or more (i.e. repitition of equally colored fields is allowed).

Up to four colors, I figured that a solution would just be projections of a tetrahedron. For these solution you need only exactly $n$ fields and if you want them to they are even finite or "compact" as board:

But from there on it gets tricky.

For which $n$ is it possible to solve the problem described above and why?

If there are some ways around it, how can the constructions be formalized?

Sadly, I guess there are some topological restrictions above $n=4$. This leads to a more difficult secondary follow up question. Let's say I come up with a plane, which doesn't fulfill the criteria of equal propability for each color, like for example in

where there are some red fields, which don't connect to blue ones. Then if I start somewhere and take $100$ random steps, I will collect fewer blue than yellow steps.

Is there a canonical way to classify the dependency of the descriptive statistics of the random walk on the structure and coloring of the plane?

This gets difficult very fast, but I can imagine inductive investigations on geometries with high symmetry.

I also have a little second question for the dice metaphor, which has no definitive answer and is too trivial for a seperate post:

To define a "macroscopic property" of a dice roll, what are some good/interesting ways to distinguish two out of the six numbers $\{1,2,3,4,5,6\}$? The best thing I could come up with is "these numbers are divisible by 3", but that's kind of lame.

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On the last question: odd primes, powers of $3$, powers of $4$, powers of $5$, powers of $6$, positive powers of $2$, perfect squares, ... – joriki Apr 16 '12 at 10:01
@joriki: "odd primes" Haha. Yes, I like that one. – NikolajK Apr 16 '12 at 10:03

For $n=7$ you can color a hexagonal tiling of the plane, see here for a picture. (This refers to the "chromatic number of the plane".) There are tilings of the hyperbolic plane with regular $n$-gons of a certain size for every $n\geq 7$. Looking at such figures might give you an idea for finding some "systematic" coloring of such a tiling.