Inspired by the upcoming book 10 PRINT CHR$(205.5+RND(1)); : GOTO 10 by Nick Montfort et al., whose title derives from this particular example of emergent behavior.
Here's an example:
(Note that I'm considering the grey "negative space" here, not the graph made by the black "walls.")
In particular, this breaks an rectangle up into a number of disjoint paths. What can be said about the length of such paths, say asymptotically as $m, n \to \infty$ while $m / n$ remains constant? (To make things a bit cleaner, it may make more sense to identify the edges to make a torus instead of leaving the edges ragged.
We can get some initial results just by looking at Euler characteristic, as described by Anders Kaseorg at the link below
See also the discussion here: http://www.quora.com/Combinatorics/Take-an-m-x-n-piece-of-grid-paper-and-in-each-box-pick-two-opposite-corners-at-random-to-connect-What-can-be-said-about-the-resulting-pattern
Example in UNICODE if you can't see the picture:
╱╱╲╱╲╲╱╲╱╱╲╲╱╱╱╲╱╱╱╱╱╱╱╱╲╱╱╲╲╱╲╱╱╱╱╲╱╱╲ ╲╱╲╱╱╱╲╱╲╲╱╲╲╱╲╱╱╲╱╲╲╲╲╱╱╱╲╱╲╲╲╲╱╲╲╲╲╲╲ ╲╱╲╱╲╲╲╲╱╲╱╲╲╱╲╲╲╲╱╲╲╱╲╲╱╲╱╱╲╱╲╱╲╱╲╱╲╱╲╱ ╲╱╲╱╲╲╲╲╱╱╲╲╲╱╱╲╲╲╲╲╲╱╱╱╱╲╲╱╲╱╱╱╱╱╲╱╱╱╱╱ ╱╱╱╱╲╱╱╱╲╲╲╲╲╲╲╲╲╱╱╱╲╱╲╲╲╱╲╲╱╲╲╲╲╲╲╱╲╱╱╱ ╱╱╲╲╱╱╱╲╲╲╱╲╱╱╱╱╲╱╱╱╱╱╲╱╲╲╱╱╱╱╲╱╲╲╱╲╱╲╱╱ ╲╱╱╱╱╲╱╲╱╱╲╱╱╲╱╱╲╱╲╲╱╲╱╲╲╲╱╱╱╲╲╱╱╲╱╲╲╱╲╲ ╲╱╲╱╱╱╲╲╱╱╲╲╲╱╱╱╲╲╱╲╲╱╱╱╲╲╱╲╱╲╲╲╱╲╱╲╲╱╱╲╱ ╲╲╱╲╱╱╲╱╲╱╲╱╱╱╱╱╱╲╱╲╱╱╱╱╲╱╲╱╱╲╲╲╱╲╲╲╱╱╲╲╱ ╲╱╱╲╲╲╲╲╲╱╲╲╲╲╲╲╱╲╱╱╱╲╱╲╱╱╱╲╲╲╲╲╱╲╲╲╱╱╲╲╲ ╱╲╱╲╱╲╱╲╱╲╱╲╲╱╲╲╲╱╱╲╲╲╲╲╲╲╱╱╲╱╲╲╲╱╲╱╲╱╲╲╲ ╱╱╱╱╱╲╲╱╲╲╲╱╱╲╲╲╲╱╱╱╲╱╲╱╲╱╲╱╲╲╲╱╲╲╲╱╲╲╱╱╱
(Thanks to Alon Amit for the sample picture.)