Mathematisches Kolloquium

Zusammenfassung: We introduce and study a universal model of random geometry in two dimensions. To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class of graphs with a given size $n$, for instance the class of all triangulations of the sphere with $n$ faces. We equip the vertex s et of the graph with the usual graph distance rescaled by the factor $n^{-1/4}$. We then prove that the resulting random metric space converges in distribution as $n\to\infty$, in the Gromov-Hausdorff sense, toward a limiting random compact metric space called the Brownian map, which is universal in the sense that it does not depend on the class of graphs chosen initially. The Brownian map is homeomorphic to the sphere, but its Hausdorff dimension is equal to $4$. We obtain detailed information about the structure of geodesics in the Brownian map. This study is motivated in part by the use of random geometry in the physical theory of two-dimensional quantum gravity

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