NONBIJECTIVE SCALING LIMIT OF MAPS VIA RESTRICTION

The main purpose of this work is to provide a framework for proving that, given a family of random maps known to converge in the Gromov–Hausdorff sense, then some (suitable) conditional families of random maps converge to the same limit. As a proof of concept, we show that quadrangulations with a simple boundary converge to the Brownian disk. More precisely, we fix a sequence (pn) of even positive integers with pn ∼ 2α √ 2n for some α ∈ (0,∞). Then, for the Gromov–Hausdorff topology, a quadrangulation with a simple boundary uniformly sampled among those with n inner faces and boundary length pn weakly converges, in the usual scaling n−1/4, toward the Brownian disk of perimeter 3α.

[1]  W. D. Kaigh.  An Invariance Principle for Random Walk Conditioned by a Late Return to Zero , 1976 .

[2]  Philippe Flajolet,et al.  Random maps, coalescing saddles, singularity analysis, and Airy phenomena , 2001, Random Struct. Algorithms.

[3]  Elton P. Hsu,et al.  THE SCALING LIMIT OF RANDOM SIMPLE TRIANGULATIONS AND RANDOM SIMPLE QUADRANGULATIONS BY , 2017 .

[4]  Luis Fredes,et al.  Tree-Decorated Planar Maps , 2020, Electron. J. Comb..

[5]  Thomas Leh'ericy.  First-passage percolation in random planar maps and Tutte’s bijection , 2019, Electronic Journal of Probability.

[6]  Emmanuel Jacob,et al.  The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection , 2013, 1312.5842.

[7]  L. Addario-Berry,et al.  Joint convergence of random quadrangulations and their cores , 2015, 1503.06738.

[8]  Donald L. Iglehart,et al.  Weak Convergence to Brownian Meander and Brownian Excursion , 1977 .

[9]  Gr'egory Miermont,et al.  The Brownian map is the scaling limit of uniform random plane quadrangulations , 2011, 1104.1606.

[10]  The Brownian disk viewed from a boundary point. , 2020, 2006.11050.

[11]  Nicolas Curien,et al.  A view from infinity of the uniform infinite planar quadrangulation , 2012, 1201.1052.

[12]  C. Abraham.  Rescaled bipartite planar maps converge to the Brownian map , 2013, 1312.5959.

[13]  Philippe Di Francesco,et al.  Planar Maps as Labeled Mobiles , 2004, Electron. J. Comb..

[14]  L. Addario-Berry,et al.  Convergence of odd-angulations via symmetrization of labeled trees , 2019, 1904.04786.

[15]  Nicolas Curien,et al.  Uniform infinite planar quadrangulations with a boundary , 2012, Random Struct. Algorithms.

[16]  J. L. Gall,et al.  First-passage percolation and local modifications of distances in random triangulations , 2015, Annales scientifiques de l'École normale supérieure.

[17]  Jason Miller,et al.  Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk , 2017, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[18]  A. Shiryayev.  On Sums of Independent Random Variables , 1992 .

[19]  Scaling limit of large triangulations of polygons , 2019, 1910.04946.

[20]  Jérémie Bettinelli,et al.  Scaling Limits for Random Quadrangulations of Positive Genus , 2010, 1002.3682.

[21]  J. Bouttier,et al.  Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop , 2009, 0906.4892.

[22]  J. Gall.  Itô’s excursion theory and random trees , 2010 .

[23]  Ewain Gwynne,et al.  External diffusion-limited aggregation on a spanning-tree-weighted random planar map , 2019, 1901.06860.

[24]  R. Cori,et al.  Planar Maps are Well Labeled Trees , 1981, Canadian Journal of Mathematics.

[25]  Jérémie Bettinelli.  Scaling Limit of Random Planar Quadrangulations with a Boundary , 2011, 1111.7227.

[26]  Jean-Franccois Le Gall,et al.  Uniqueness and universality of the Brownian map , 2011, 1105.4842.

[27]  I. Kortchemski.  A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees , 2011, 1109.4138.