First, consider the two-dimensional lattice Z^2. Let each point of it be a "tunnel", a "\" mirror, a "/" mirror or a "normal point" independently from the others. Once a realization of this random field is fixed, a particle performing a random walk on Z^2 given this realization obeys the following laws: If a point the walker hits is a tunnel, it goes straight through it; a mirror, it reflects from it like a beam of light would; a normal point, it behaves like a simple RW, i.e. choses one of the four possible neighbors with equal probabilities.

This model is called a "random walk in a random labyrinth" and was introduced by Grimmett, Menshikov and V. in 1996.

The labyrinth is called {\em recurrent} if the RW on it is recurrent, and transient otherwise. The labyrinth is called {\em localised} if the number of points which a walker can visit is finite.

Assuming that the density of normal points is non-zero, we show 1) in the case of Z^2, a labyrinth is recurrent a.s. and 2) under which conditions it is non-localized with positive probability. For a general model on Z^d, d>2, we present sufficient conditions ensuring that a labyrinth is transient with positive probability.