Growth and percolation on the uniform infinite planar triangulation

By Omer Angel. 2002

190 citations recorded by the source corpus.

AbstractA construction as a growth process for sampling of the uniform in- finite planar triangulation (UIPT), defined in [AnS], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT.By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r4 up to polylogarithmic factors, in accordance with heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r2 up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an m-gon (also defined in [AnS]) converges in distribution to an asymmetric stable random variable of type 1/2.By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability pc = 1/2 and that at pc percolation does not occur.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a0455912ce7b8dba6570cf9642f6bacb7d1ca6e9

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Citations in the corpus

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.”
Local limit of labeled trees and expected volume growth in a random quadrangulation (2003, background): “In [7] it is then proven that the ball of radius r around a marked triangle in a sample triangulation contains on the order of r4 triangles up to logarithmic corrections, which can be seen as yet another manifestation of the exponent α being equal to 4.”
The Brownian Plane (2012, background): “Such local limits were studied first in the case of triangulations by Angel and Schramm [3, 4] and the limiting object is called the uniform infinite planar triangulation (UIPT).”
Recurrence of planar graph limits (2012, background): “Angel [4] showed that a ball of radius r has volume r4+o(1) and the boundary component, separating this ball from infinity, has size r2+o(1) (see also [12]).”
Processes on Unimodular Random Networks (2006, background): “(Thus, such graphs do not include regular trees or hyperbolic graphs, other than trivial examples like Z.) A specialization to random weak limits of plane triangulations was studied in more detail in interesting recent work of Angel and Schramm (2003) and Angel (2003).”
Local structure of random quadrangulations (2005, background): “They also provided estimates for the growth of hull boundary, which was shown to be quadratic in radius with polylog corrections [3], but no exact limit was found.”
Coarse geometry and randomness (2013, background): “I would like to deeply thank Omer Angel, Louigi Addario-Berry, Agelos Georgakopoulos, Vladimir Shchur for comments, remarks and corrections, and Nicolas Curien, Ron Rosenthal and Johan Tykesson for great help with editing, collecting and joining together the material presented. Some of the proofs wi”
Percolation on uniform infinite planar maps (2013, methodology): “We then describe in Section 3 a “peeling process” similar to the process introduced by Angel in [3] for triangulations.”
Planar stochastic hyperbolic triangulations (2016, methodology): “The UIPT and its quadrangular analog the UIPQ have received a lot of attention in recent years [4, 12, 22, 27, 31] partially motivated by the connections with the physics theory of 2dimensional quantum gravity, the Gaussian free field [19] and the Brownian map [20, 30, 34].”
The hull process of the Brownian plane (2014, result): “This heuristic discussion is made much more precise in the companion paper [10], where many of the results of the present work are interpreted in terms of asymptotics for the so-called “peeling process” studied by Angel [4] for the UIPT.”
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): “L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.”
A view from infinity of the uniform infinite planar quadrangulation (2012, result): “This approach allows us to prove the conjectures of Krikun [21] pertaining to the “geometry at infinity” of the UIPQ, and to derive new results about the UIPQ, among which a fine study of infinite geodesics.”
Random geometry on the sphere (2014, methodology): “In particular, Angel [6] has studied percolation on the UIPT.”
Conformally invariant scaling limits: an overview and a collection of problems (2006, background): “To date, there has been progress in the mathematical (as well as physical) understanding of the statistical physics in the plane as well as in quantum gravity [2], [3], [16].”