Random planar lattices and integrated superBrownian excursion

By Philippe Chassaing, Gilles Schaeffer. 2002

239 citations recorded by the source corpus.

Abstract.In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r=R−L of the support of the one-dimensional ISE, or precisely: More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half-line, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks to the Brownian snake description of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in two-dimensional pure quantum gravity.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/216bc3a14c7f59e1167691080a0d30b5c5bfc4ba

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Citations in the corpus

Scaling limits of random trees and planar maps (2011, methodology): “The points (i) and (iii) are due to Chassaing and Schaeffer [12], and (ii) is due to Le Gall [30], although these references state these properties in a slightly different context, namely, in the case where v∗ is the root vertex rather than a uniformly chosen vertex.”
Uniqueness and universality of the Brownian map (2011, methodology): “To complete this introduction, let us mention that the idea of studying the continuous limit of large random quadrangulations first appeared in the pioneering paper of Chassaing and Schaeffer [7], which obtained detailed information about the asymptotics of distances from the root vertex.”
Local limit of labeled trees and expected volume growth in a random quadrangulation (2003, methodology): “A different method was recently applied in [ 12 ] to quadrangulated planar maps, making it possible to exploit a clever bijective correspondence between such maps and so-called well-labeled trees.”
Conditioned Brownian trees (2005, background): “orem 2.3). This result is clearly related to the invariance of the CRT under uniform re-rooting, which was observed by Aldous [4] (and generalized to L´evy trees in Proposition 4.8 of [13]). See also [9] for similar ideas in a discrete setting, and especially Proposition 13 of [26] which gives a clo”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, background): “In the case of planar quadrangulations, the bijection in question is the so-called Cori–Vauquelin–Schaeffer bijection [CV81, Sch98, CS04] between planar quadrangulations and so-called well-labeled trees.”
An invariance principle for random planar maps (2006, unknown): “Marckert and Miermont (2006) show that similar results as in Chassaing and Schaeffer (2004); Le Gall (2006); Marckert and Mokkadem (2006) hold for rooted and pointed bipartite random planar maps (whose faces all have even degree) taken under a Boltzmann-type distribution, where each face of degree…”
The Brownian Plane (2012, result): “We will describe Schaeffer’s bijection between Tn × {0, 1} and the set of all rooted and pointed quadrangulations with n faces.”
The hull process of the Brownian plane (2014, methodology): “We then generate the edges of Q∞ by the following device, which is analogous to the Schaeffer bijection between finite (rooted) quadrangulations and well-labeled trees [8].”
The topological structure of scaling limits of large planar maps (2006, background): “it seems plausible that scaling limits of large bipartite planar maps can be described in terms of continuous random trees. This idea already appeared in the pioneering work of Chassaing and Schaeﬀer [12], and was then developed by Marckert and Mokkadem [26], who deﬁned and studied the so-called Bro”
Limit of normalized quadrangulations: The Brownian map (2004, background): “Theyarealsoequaltothelim itradiuswhen ! Q n isendowed with theuniform distribution with the sam enorm alization (see [15]).”
On the sphericity of scaling limits of random planar quadrangulations (2007, background): “We let Qn be the set of rooted(2) quadrangulations of the sphere with n faces, which is a finite set of cardinality 2 · 3(2n)!/(n!(n+ 2)!), see [5].”
Combinatorial Stochastic Processes (2006, methodology): “• random partitions related to uniformly chosen invertible matrices over a finite field, as studied by Fulman [160]; • random maps, coalescing saddles, singularity analysis, and Airy phenomena, [81]; • random planar lattices and integrated superbrownian excursion [94].”
Rational and algebraic series in combinatorial enumeration (2008, unknown): “This has led to remarkable connections with a random probability distribution called the Integrated SuperBrownian Excursion [32].”
A view from infinity of the uniform infinite planar quadrangulation (2012, background): “The rooting of τ is also recovered from q and ` by the same procedure as in the finite case.”
Scaling limits of random planar maps with large faces (2011, result): “ﬁnity (or equivalently when the number of vertices tends to inﬁnity), does not depend on the choice of pand is given by the so-called Brownian map. Since the pioneering work of Chassaing and Schaeﬀer [6], there has been several results supporting this conjecture. Marckert and Mokkadem [22] introduce”