Local limit of labeled trees and expected volume growth in a random quadrangulation

By Philippe Chassaing, Bergfinnur Durhuus. 2003

109 citations recorded by the source corpus.

Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton–Watson trees. As a consequence, we find that the expected volume of the ball of radius r around a marked point in the limit random surface is Θ(r4).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9668fbd58563c04138675531479a2f296cfba6d9

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Planar stochastic hyperbolic triangulations (2016, unknown): “[3, 5, 15, 17, 18, 29] and the references therein.”
The hull process of the Brownian plane (2014, methodology): “The resulting collection of edges forms an infinite quadrangulation of the plane, which is the UIPQ Q∞ (see Fig.”
Random geometry on the sphere (2014, background): “A completely different construction of M (4) ∞ based on a version of the CVS bijection for infinite trees was given in [17] (see [38] for a proof of the fact that the two constructions give the same object).”
The topological structure of scaling limits of large planar maps (2006, background): “See Bouttier’s thesis [7] and the references therein for applications of these bijections to the statistical physics of random surfaces.”
A view from infinity of the uniform infinite planar quadrangulation (2012, methodology): “Details can be found in [11], see also [27, 30].”
Some families of increasing planar maps (2007, unknown): “asure on triangulations with nfaces to a measure on the set of inﬁnite but locally ﬁnite rooted triangulations (see also Krikun [26] for a simple description of this measure). Chassaing & Durhuus [12] obtained then a similar result, with a totally diﬀerent approach, on uniform rooted quadrangula”
Percolation on uniform infinite planar maps (2013, background): “Both the UIPT and the UIPQ have been the focus of numerous works in recent years such as [3, 9, 12, 15, 24, 25], but it is fair to say that they are not yet fully understood.”
Scaling limits for the peeling process on random maps (2014, methodology): “Our methods also apply to the Uniform Infinite Planar Quadrangulation (UIPQ), which was constructed independently by Krikun [21] and by Chassaing and Durhuus [13] (the equivalence between these two constructions was obtained by Ménard [25]).”
Scaling limits of random planar maps with a unique large face (2012, background): “Angel and Schramm [3] studied local convergence in the case of uniformly distributed triangulations (all faces have degree 3) and later Durhuus and Chassaing [16] and Krikun [33] studied the case of uniformly distributed quadrangulations (all faces have degree 4).”
Local structure of random quadrangulations (2005, background): “ided estimates for the growth of hull boundary, which was shown to be quadratic in radius with polylog corrections [6], but no exact limit was found. A similartheorem wasprovedby Chassaingand Durhuus [8], who showedthe local weak convergence of well-labeled trees, which are known to be bijective to ”
Geodesics in large planar maps and in the Brownian map (2008, background): “In a different but related direction, the papers of Angel and Schramm [5], Angel [4] and Chassaing and Durhuus [12] study various properties of random infinite planar maps that are uniformly distributed in some sense.”
An invariance principle for random planar maps (2006, unknown): “Marckert and Mokkadem (2006) show that a slightly different family of random quadrangulations with n faces (pointed, and uniformly rooted on an edge adjacent to the distinguished vertex) converges in a loose sense, once rescaled by the factor n, to a random metric space they call the Brownian map, w”
The Brownian Plane (2012, background): “in [9] (note that a different version of the construction in the infinite case had appeared earlier in [7]).”
Limit of normalized quadrangulations: The Brownian map (2004, background): “W e endow T with the m etric dT((f1; 1);(f2; 2))= kf1 f2k1 + k 1 2k1 : Considerthe T-valued random variable (r;v)whose distribution follows: (v(t))t2[0;1] law = p 2(e(t))t2[0;1],wheree isthenorm alized Brownian excur- sion.”