Scaling Limits for Random Quadrangulations of Positive Genus

By Jérémie Bettinelli. 2010

49 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a2c67a0d6367edda1130654451e6c56d53ad78f

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, background): “We refer to [2] for a recent and quite complete treatment of first-passage bridges and limit theorems for their discrete versions, which will be helpful to us later on.”
Scaling limits of random planar maps with a unique large face (2012, unknown): “The result now follows by a conditional version of Donsker’s invariance theorem; see, for example, [10], Lemma 10, for a detailed proof.”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, methodology): “One of the main difficulties that was not present in [7,8] arises from the fact that the Brownian labels on the continuum random forest we construct do not have the same diffusion factor on the floor than in the trees.”
Compact Brownian surfaces I: Brownian disks (2015, background): “nf 0vuX v isthepastinﬁmumofX. NoteinparticularthatZ 0 s andZ 0 s0 are independentifs,s0belongtotwodiﬀerentexcursionintervalsofXaboveX. Itisclassical [LG99]thatZ0 admitsacontinuousmodiﬁcation,seealso[Bet10]foradiscussioninthe current context. For this modiﬁcation, we a.s. have Z0 Ty = 0 for every y2[”
Scaling limits of random planar maps with large faces (2011, background): “Such results are part of the folklore of the subject; see Lemma 10 in [3] for a detailed proof of a more general statement.”