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Scaling limits for random quadrangulations of positive genus. (English) Zbl 1226.60047
Summary: We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given \(g\), we consider, for every positive integer \(n\), a random quadrangulation \(q_n\) uniformly distributed over the set of all rooted bipartite quadrangulations of genus \(g\) with \(n\) faces. We view it as a metric space 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, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to 4. Our main tool is a bijection introduced by G. Chapuy, M. Marcus and G. Schaeffer [SIAM J. Discrete Math. 23, No. 3, 1587–1611 (2009; Zbl 1207.05087)] between the quadrangulations we consider and objects they call well-labeled \(g\)-trees. An important part of our study consists in determining the scaling limits of the latter.

60F17 Functional limit theorems; invariance principles
60D05 Geometric probability and stochastic geometry
Zbl 1207.05087
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