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We introduce and study a non-oriented first passage percolation model havinga property of statistical invariance by time reversal. This model is defined ina graph having directed edges and the passage times associated with each set ofoutgoing edges from a given vertex are distributed according to a generalizedBernoulli-Exponential law and i.i.d. among vertices. We derive the statisticalinvariance property by time reversal through a zero-temperature limit of therandom walk in Dirichlet environment model.

A non-oriented first passage percolation model and statistical invariance by time reversal