Random Walks in Random Labyrinths

G.R. Grimmett, M.V. Menshikov, S.E. Volkov

1996, v.2, №1, 69-86


A random labyrinth is a disordered environment of scatterers on a lattice. A beam of light travels through the medium, and is reflected off the scatterers. The set of illuminated vertices is studied, under the assumption that there is a positive density of points, called `normal points', at which the light behaves in the manner of a simple symmetric random walk. The ensuing `random walk in a labyrinth' is found to be recurrent in two dimensions, and also non-localised under certain extra assumptions on the underlying probability distribution. The walk is shown to be transient (with strictly positive probability) in three and higher dimensions, subject to the assumption that the density of `non-trivial' scatterers is sufficiently small. The principal arguments used in deriving such results originate in percolation theory. In addition, we utilise the relationship between random walks and electrical networks, namely that a random walk is recurrent if and only if a certain electrical network has infinite resistance.

Keywords: random walk,random environment,random labyrinth,scatterer,mirror,percolation,electrical network


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