Reversible Poisson-Kirchhoff Systems.

We define a general class of random systems of horizontal and vertical broken lines on the quarter plane whose distribution is proved to be translation invariant. This invariance follows from a reversibility property of the model. This class of systems generalizes several classical processes of the same kind, such as Hammersley's broken line processes involved in Last Passage Percolation theory or such as the six-vertex model for some special sets of parameters. The novelty comes here from the introduction of an intensity associated with each line. The lines are initially generated by some spatially homogeneous weighted Poisson Point Process and their evolutions (turn, split, coalescence, annihilation) are ruled by a Markovian dynamic which preserves Kirchhoff's node law for the line intensities at each intersection.