Soft question: mathematics about truchet tiles

Question

It seems that this is the first question on Truchet tiles on MO.

Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.

I wonder if there may be some interesting and deep math behind random Truchet tiles, but I do not know much about this topic. As I can guess, there might be questions like these:

1. Given a random tile in finite grid, what is the expectation length of a closed curve?

2. Questions like percolation on this graph, for example, give an infinite tiling of the plane, what is the probability that there is an infinite connected area? (2-color the basic pattern with blue and red, then the red area is like cells.)

3. Connections with the $O(1)$ loop model. (I'm sorry, I know very little about this, but I do think it is in the picture.)

Of course this list is very incomplete, and I hope someone can fill it and show me the connections with other branches of mathematics. Thanks!

Answer 1

To answer your question 3:

Indeed, in statistical mechanics such a model is known as a "loop model". In general an "O(n) loop model" (this is not the place to rant on why this is a misnomer especially in the present context) means instead of drawing truchet tile/loop configurations equiprobably, you assign them a probability which is proportional to n^(number of closed loops), where n is a fixed parameter (not necessarily an integer). These loop models have become very popular in the 80s, and are a subject of active research up to now.

The particular model corresponding to Truchet tiles is sometimes known as Temperley--Lieb loop model, sometimes also "CPL" (Completely Packed Loops). It has the nice property of being exactly solvable, in the sense of quantum integrability, which means various quantities can be computed exactly as the size of the sample goes to infinity. If you further specialize the loop weight to be 1, then this model has even more remarkable properties, allowing exact finite-size results, and interesting connections to combinatorics.

Concerning questions 1 & 2, the model in question is critical, which means such questions have a definite answer in the limit of infinite size, where they're governed by critical exponents. (which I don't know off the top of my head). In some cases, such scaling behaviors have been proven rigorously using SLE; I don't think it is so for CPL, so from a mathematical standpoint I suppose they're only "conjectures".

Ref: the best I can think of is Bernard Nienhuis' lectures "Loop models" in "Les Houches 2008: Exact methods in low-dimensional statistical physics and quantum computing", Eds Jacobsen, Ouvry, Pasquier, Serban, Cugliandolo. (you can also check out my lectures in the same volume, as well as my HDR on my webpage)