Scaling Limit of Random Planar Quadrangulations with a Boundary

By Jérémie Bettinelli. 2011

80 citations recorded by the source corpus.

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence $(\sigma_n)$ of integers such that $\sigma_n/\sqrt{2n}$ tends to some $\sigma\in[0,\infty]$. For every $n \ge 1$, we call $q_n$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2\sigma_n$ half-edges on the boundary. For $\sigma\in (0,\infty)$, we view $q_n$ as a metric space by endowing its set of vertices with the graph metric, rescaled by $n^{-1/4}$. We show that this metric space converges in distribution, at least along some subsequence, toward a limiting random metric space, in the sense of the Gromov--Hausdorff topology. We show that the limiting metric space is almost surely a space of Hausdorff dimension 4 with a boundary of Hausdorff dimension 2 that is homeomorphic to the two-dimensional disc. For $\sigma=0$, the same convergence holds without extraction and the limit is the so-called Brownian map. For $\sigma=\infty$, the proper scaling becomes $\sigma_n^{-1/2}$ and we obtain a convergence toward Aldous's CRT.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a2ff364a83810042626986b02a66d3dc0d842731

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Citations in the corpus

Scaling Limits of Random Graphs from Subcritical Classes (2014, background): “ng object of many dierent classes of discrete structures, in particular trees, see e.g. Haas and Miermont [25] and references therein, and planar maps, see e.g. Albenque and Marckert [2], Bettinelli [9], Caraceni [12], Curien, Haas and Kortchemski [14] and Janson and Stefansson [28]. Although the af”
Scaling limits of random planar maps with a unique large face (2012, background): “Bettinelli [9] showed a similar convergence of uniform quadrangulations with a boundary toward Aldous’ continuum random tree when the length of the boundary grows sufficiently fast and the distances in the quadrangulations are divided by the square root of the length of the boundary (see also the wo”