I have been simulating some percolation processes. Here, one starts with the 2-dimensional lattice Z^2 and at each point, one puts a two-sided mirror (independently with probability p) or no mirror (with probability 1-p). The mirrors are at 45 degrees to the axes (so may be represented as / and \). The two orientations of the mirrors are equiprobable and the probabilities at different sites are independent. One then shines a beam of light from the origin in one of the 4 axial directions and follows it as it deflected by the mirrors. This is illustrated here .

Either the light forms a finite loop or it goes out to infinity. In the former case, the light is called localized. For p=1, all light beams are known to be localized with probability 1. The aim is to show whether the light is localized with probability 1 for different values of p. Here is a simulation of 10^7 times for p=0.8 in which the light has not yet returned. Will it come back eventually?

The blue (I hope you're looking at this in colour) represents the forward path of the light beam and the red the backwards. The green parts are where the forward and backward paths have entered the same cell (about 50x50 lattice sites).

For detailed information about this problem, read the paper .

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