Scaling Limit of Arbitrary Genus Random Maps

Abstract : In this work, we discuss the scaling limits of two particular classes of maps. In a first time, we address bipartite quadrangulations of fixed positive genus g and, in a second time, planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We view these objects as metric spaces by endowing their sets of vertices with the graph metric, suitably rescaled.We show that a map uniformly chosen among the maps having n faces in one of these two classes converges in distribution, at least along some subsequence, toward a limiting random metric space as n tends to infinity. This convergence holds in the sense of the Gromov--Hausdorff topology on compact metric spaces. We moreover have the following information on the limiting space. In the first case, it is almost surely a space of Hausdorff dimension 4 that is homeomorphic to the genus g surface. In the second case, it is almost surely a space of Hausdorff dimension 4 with a boundary of Hausdorff dimension 2 that is homeomorphic to the unit disc of R^2. We also show that in the second case, if the length of the boundary is little-o of the square root of the number of faces, the same convergence holds without extraction and the limit is the same as for quadrangulations without boundary, that is the Brownian map.
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Jérémie Bettinelli. Scaling Limit of Arbitrary Genus Random Maps. General Mathematics [math.GM]. Université Paris Sud - Paris XI, 2011. English. ⟨NNT : 2011PA112213⟩. ⟨tel-00638065v2⟩

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