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In a rst-passage percolation model on the square lattice 2, if the passage times are inde- pendent then the number of geodesics is either 0 or +1. If the passage times are stationary, ergodic and have a nite moment of order >1=2, then the number of geodesics is either 0 or +1. We construct a model with stationary passage times such that E(t(e)) < 1, for every 0 < <1=2, and with a unique geodesic. The recurrence/transience properties of reversible random walks in a random environment with stationary conductances (a(e); e is an edge of 2) are considered.

Geodesics and Recurrence of Random Walks in Disordered Systems