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Abstract: We discuss scaling limits of large bipartite quadrangulations of positivegenus. For a given $g$, we consider, for every $n \ge 1$, a randomquadrangulation $\q n$ uniformly distributed over the set of all rootedbipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metricspace by endowing its set of vertices with the graph distance. We show that, as$n$ tends to infinity, this metric space, with distances rescaled by the factor$n^{-1-4}$, converges in distribution, at least along some subsequence, towarda limiting random metric space. This convergence holds in the sense of theGromov-Hausdorff topology on compact metric spaces. We show that, regardless ofthe choice of the subsequence, the Hausdorff dimension of the limiting space isalmost surely equal to 4. Our main tool is a bijection introduced by Chapuy,Marcus, and Schaeffer between the quadrangulations we consider and objects theycall well-labeled $g$-trees. An important part of our study consists indetermining the scaling limits of the latter.



Author: Jérémie Bettinelli LM-Orsay

Source: https://arxiv.org/



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