This is inspired by the illustration in this recently updated question. So we take a (fairly big) $n$ and an $n \times n$ grid where we draw at random one diagonal in each of the $1 \times 1$ squares.

The result is a kind of maze in the 45° rotated grid of $\sqrt2 \times \sqrt2$ squares (which I'll now consider as new unit cells $-$ more precisely, the grid cells would be in fact $\frac{\sqrt2}2 \times \frac{\sqrt2}2 $, as the width of the labyrinth paths is $\frac{\sqrt2}2 $). I was wondering about the sizes of the connected components of such a labyrinth, and so I took Joseph O'Rourke's sample illustration and colored everything bigger than $1 \times 1$ and $1 \times 2$, which yields the following outcome:

For the two light blue regions and the big pink one between them, I have cheated a bit in coloring, as all three have tiny "leaks" into the infinite (yellow) part outside. In a strict sense, those three big regions would have to be yellow, but I think it makes sense to consider the infinite part as essentially linked to boundary effects only.

Asking about the average area of a connected region for big $n$ may thus be an ill-defined question, and even if we only consider the "completely interior" components (i. e. here all red, green, dark blue and white ones), I guess that such a question is way too hard to be feasible. But the following should be rather easy, as it only concerns local neighborhoods:

- What is the average proportion of components consisting of a $1 \times 1$ cell (not among all components, rather as a fraction of $n^2$, neglecting boundary effects) ?
- Same question for $1 \times 2$ components.

Of course, the probability to obtain such a unit component inside a given $2 \times 2$ subsquare of the original grid is simply $\frac1{16}$, but we cannot conclude from this that there are a total of $\frac{n^2}{32}$ (or $\frac{n^2}{16}$?) of them in average, as the $2 \times 2$ subsquares of the original grid can overlap. It may just need a simple inclusion-exclusion approach, but I don't see how, especially for the $1 \times 2$ components.