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On the enumeration of random maps with three tight boundaries and its probabilistic consequences

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We consider the enumeration problem of graphs on surfaces, or maps, with three boundaries, also colloquially called pairs of pants. Perhaps surprisingly, a formula due to Eynard and extended by Collet-Fusy shows that this problem has a very simple and explicit solution, which becomes even simpler when one asks that the boundaries are tight, meaning that they have smallest possible length in their free homotopy class. We provide a bijective approach to this formula which consists in decomposing the graph into elementary pieces in a way that is reminiscent of certain geometric constructions of pairs of pants in hyperbolic geometry. I will also discuss the probabilistic consequences of this bijective approach by studying statistics of minimal separating loops in random maps with boundaries. (Based on joint work with J\’er\’emie Bouttier and Emmanuel Guitter.)

This talk is part of the Probability series.

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