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Geodesic stars in Brownian geometry

With Jean-Francois Le Gall (Universite Paris-Saclay)

Geodesic stars in Brownian geometry

Planar maps, which are finite connected graphs embedded in the sphere, are basic discrete models of random geometry and are usually chosen uniformly at random in a given class, for instance the class of all triangulations with a fixed number of faces. The so-called Brownian sphere, or Brownian map, is the random metric space obtained as the universal scaling limit of random planar maps equipped with the usual graph distance, in the Gromov-Hausdorff topology. We discuss geodesics in the Brownian sphere. It has been known for some time that any two geodesics starting from a typical point of the Brownian sphere must coincide near their starting point. However, for any m < 5, there are exceptional points called geodesic stars with m arms, which are starting points of m disjoint geodesics. We prove that the Hausdorff dimension of geodesic stars with m arms is equal to 5-m . This complements an earlier work of Miller and Qian who proved that this Hausdorff dimension is bounded above by 5-m.

The talk will be held online using zoom. The link will be distributed to the probability seminar list. If you are not on the list and would like to attend the talk, please email Perla Sousi (ps422@cam.ac.uk) for the link.

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