I'm interested in building maze-like structures on the [5, 4] tiling of the hyperbolic plane, where by maze-like I mean something akin to a spanning tree of the underlying lattice: a subgraph of the lattice that's still connected (every cell can reach every other cell) but with a sort of 'distance multiplier' effect where from any given cell in the lattice, there are likely to be cells that are nearby in the original tiling with a long 'shortest path' through the subgraph to reach that point. (This is an admittedly imprecise notion, but I doubt there are explicit results on this sort of thing anyway). I could use any of the standard maze-generation algorithms, but they involve a 'prepass' that holds the entire lattice graph in memory and so aren't really practical for infinite (or functionally infinite) lattices; I'm looking for something that can generate walls (or equivalently, cut edges from the underlying graph) in a 'local' fashion without needing to look explicitly at the global structure.

If I were on the plane, then there are relatively straightforward ways of doing this: for instance, for each cell, place a wall randomly on either that cell's south or east edge. With probability 1 this will create a spanning tree of the plane - it's not a 'random' spanning tree (in particular, of course, it's impossible for any cell to have both its south and east sides 'walled', which I imagine happens with probability 0 for a random spanning tree of the plane) but it's a tree, even so. For instance, here are two $5\times 5$ grids of cells with their east or south sides randomly walled:

Similarly, if the generation process uses a 3-sided coin with equal probabilities of drawing a south wall, east wall or no wall, then the resulting grids aren't spanning trees (there can exist multiple disjoint paths between nodes), but they still clearly have the sort of distance multiplication I'm looking for:

Unfortunately, these techniques don't work as well for the hyperbolic grid I'm interested in; getting a canonical notion of outward edges is a bit more challenging, and the 'maziness' of the result is much harder to control. Does anyone know of a scheme similar to the coin-flipping techniques with only local information generated/used that can be applied to arbitrary regular tilings?