Abstract
Abstract. 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 $n$ to the power of $-1/4$, 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 Chapuy, Marcus, and Schaeffer 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.
Highlights
1.1 MotivationThe aim of the present work is to investigate scaling limits for random maps of arbitrary genus
Recall that a map is a cellular embedding of a finite graph into a compact connected orientable surface without boundary, considered up to orientationpreserving homeomorphisms
We will focus on bipartite quadrangulations: a map is a quadrangulation if all its faces have degree 4; it is bipartite if each vertex can be colored in black or white, in such a way that no edge links two vertices that have the same color
Summary
The aim of the present work is to investigate scaling limits for random maps of arbitrary genus. Using bijective approaches developed by Cori and Vauquelin [8] between planar quadrangulations and so-called well-labeled trees, Chassaing and Schaeffer [7] exhibited a scaling limit for some functionals of a uniform random planar quadrangulation They studied in particular the so-called profile of the map, which records the number of vertices located at every possible distance from the root, as well as its radius, defined as the maximal distance from the root to a vertex. The present work generalizes a part of the above results to any positive genus: we will show the tightness of the laws of rescaled uniform random bipartite quadrangulations of genus g with n faces in the sense of the Gromov-Hausdorff topology. We will prove that the Hausdorff dimension of every possible limiting space is almost surely 4
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