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Geodesic distance in planar maps: from matrix models to trees
Matrix models have proved to be a successful tool for enumerating maps of arbitrary genus, perused by physicists in the context of two-dimensional quantum gravity. We show how planar map results may be rephrased into tree combinatorics, and how they can be generalized to study refined properties such as geodesic distances within maps. Remarkably, the problem is still integrable (and exactly solvable), and shows striking similarities with the orthogonal polynomial solution of the one-matrix model.
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