Invariance principles for random bipartite planar maps

It is conjectured in the Physics literature that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing & Schaeffer, have shown that the radius of a random quadrangulation with $n$ faces converges in distribution once rescaled by $n^{1/4}$ to the diameter of the Brownian snake, up to a scaling constant. Using a bijection due to Bouttier, di Francesco \&\ Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution: the radius of such maps, conditioned to have $n$ faces (or $n$ vertices) and under a criticality assumption, converges in distribution once rescaled by $n^{1/4}$ to a scaled version of the diameter of the Brownian snake. Convergence results for the so-called profile of maps are also provided, as well as convergence of rescaled bipartite maps to the Brownian map, as introduced by Marckert & Mokkadem. The proofs of these results rely on a new invariance principle for two-type spatial Galton-Watson trees.

[1]  Thomas Duquesne,et al.  Random Trees, Levy Processes and Spatial Branching Processes , 2002 .

[2]  J. Gall.  A conditional limit theorem for tree-indexed random walk , 2006 .

[3]  Thomas Duquesne.  A limit theorem for the contour process of condidtioned Galton--Watson trees , 2003 .

[4]  D. Aldous.  Stochastic Analysis: The Continuum random tree II: an overview , 1991 .

[5]  Russell Lyons,et al.  A Conceptual Proof of the Kesten-Stigum Theorem for Multi-Type Branching Processes , 1997 .

[6]  Bergfinnur Durhuus,et al.  Quantum Geometry: A Statistical Field Theory Approach , 1997 .

[7]  Conditioned Brownian trees , 2005, math/0501066.

[8]  A Note on “State Spaces of the Snake and Its Tour—Convergence of the Discrete Snake” by J.-F. Marckert and A. Mokkadem , 2003 .

[9]  J. Pitman.  Combinatorial Stochastic Processes , 2006 .

[10]  J. L. Gall,et al.  Spatial Branching Processes, Random Snakes, and Partial Differential Equations , 1999 .

[11]  Abdelkader Mokkadem,et al.  Limit of normalized quadrangulations: The Brownian map , 2006 .

[12]  Jean-François Marckert,et al.  The depth first processes of Galton--Watson trees converge to the same Brownian excursion , 2003 .

[13]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[14]  Philippe Chassaing,et al.  Random planar lattices and integrated superBrownian excursion , 2002, math/0205226.

[15]  A. Mokkadem,et al.  States Spaces of the Snake and Its Tour—Convergence of the Discrete Snake , 2003 .

[16]  Local limit of labelled trees and expected volume growth in a random quadrangulation Philippe Chassaing , 2005 .

[17]  Omer Angel,et al.  Uniform Infinite Planar Triangulations , 2002 .

[18]  Svante Janson,et al.  Convergence of Discrete Snakes , 2005 .

[19]  David Aldous,et al.  The Continuum Random Tree III , 1991 .

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

[21]  V. Statulevičius,et al.  Limit Theorems of Probability Theory , 2000 .

[22]  D. Stroock,et al.  Probability Theory: An Analytic View. , 1995 .

[23]  Omer Angel.  Growth and percolation on the uniform infinite planar triangulation , 2002 .

[24]  V. V. Petrov,et al.  Limit Theorems of Probability Theory , 2000 .

[25]  Gilles Schaeffer.  Conjugaison d'arbres et cartes combinatoires aléatoires , 1998 .