Local structure of random quadrangulations

By Maxim Krikun. 2005

111 citations recorded by the source corpus.

This paper is an adaptation of a method used in \cite{K} to the model of random quadrangulations. We prove local weak convergence of uniform measures on quadrangulations and show that the local growth of quadrangulation is governed by certain critical time-reversed branching process and the rescaled profile converges to the reversed continuous-state branching process. As an intermediate result we derieve a biparametric generating function for certain class of quadrangulations with boundary.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/01e099640455776d28672588f0a63be68af11d0c

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In Condensed Matter Physics

In Mathematical Physics

Citations in the corpus

Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps (2014, methodology): “In the case where 2p = 4, we obtain the local convergence in distribution of large uniform quadrangulation to the UIPQ which was first obtained by Krikun in [15].”
Random geometry on the sphere (2014, background): “y also dene the limit in Theorem 5.1 as an innite planar map, that is, with a given embedding in the sphere, and this makes it possible to give a stronger form of the local convergence (see [7] and [27]). Here for simplicity we avoid dealing with innite planar maps, since only the graph structure wi”
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.”
The Brownian Plane (2012, unknown): “The analogous result for quadrangulations was obtained later by Krikun [12] (see also Chassaing and Durhuus [7] and Ménard [20]), leading to the uniform infinite planar quadrangulation (UIPQ).”
The hull process of the Brownian plane (2014, methodology): “2 have been obtained by Krikun [16, 17] in the discrete setting of the UIPT and the UIPQ.”
A view from infinity of the uniform infinite planar quadrangulation (2012, background): “Consider a proper embedding of a finite connected graph in the sphere S2 (loops and multiple edges are allowed).”
Hyperbolic and Parabolic Unimodular Random Maps (2016, background): “al spanning trees, preserve unimodularity, giving rise to many additional examples. Unimodular random maps, most notably the uniform innite planar triangulation (UIPT) [7] and quadrangulation (UIPQ) [55, 29], have also been studied in the context of 2-dimensional quantum gravity; see the survey [35]”