## Maze on a Torus

Consider a square grid on a torus. Choose a spanning tree uniformly at random. If we use this spanning tree to form walls, we obtain an interesting maze on the torus. This maze has the homology of a punctured torus, since the walls are contractible to a point. Thus, there must be some paths in the maze that represent some non-contractible cycles. How long are these cycles? There is good reason to think that on an *n* x *n* grid, their length is roughly *n*^{5/4}, but no one knows how to prove this.*- Russell Lyons*