A finite maze is generated as follows: A N×N grid of squares, where N is odd, and each border between two squares, or between a square and the outside of the N×N grid, has a p% chance of being a wall. The start is in the middle, and the goal is to get out of the N×N zone.
Let f(N,p) be the probability that there is a way out of the N×N maze with p% walls. What is the limit of f(N,p) as N goes to infinity? When is the limit 0?
Modified versions of the problem:
Walls are full squares, so squares in the grid have a p% chance to be wall tiles (and if the start is a wall tile then it is automatically inescapable).
Higher dimensional mazes.
Both.