Uniform Infinite Planar Triangulations

Angel, Omer; Schramm, Oded

doi:10.1007/s00220-003-0932-3

Cited by 174 publications

(278 citation statements)

References 32 publications

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“…Although scaling limits describe asymptotic global properties, they do not contain information on local properties, such as the limiting degree distribution of a randomly chosen vertex in a graph. Such asymptotic local properties of random structures are described by Benjamini-Schramm limits [10,13,27]. See Section 2.3 for a precise definition.…”

Section: Figurementioning

confidence: 99%

The continuum random tree is the scaling limit of unlabeled unrooted trees

Stufler

2018

Random Struct Algorithms

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We show that the uniform unlabeled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov‐Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This confirms a conjecture by Aldous (1991). We also establish Benjamini‐Schramm convergence of this model of random trees and provide a general approximation result, that allows for a transfer of a wide range of asymptotic properties of extremal and additive graph parameters from Pólya trees to unrooted trees.

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Section: Figurementioning

confidence: 99%

The continuum random tree is the scaling limit of unlabeled unrooted trees

Stufler

2018

Random Struct Algorithms

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“…It is well known that the degree of the root in the UIPT and the UIPQ has an exponential tail. See [7,Lemma 4.1 and 4.2] or [25] for the UIPT and [8,Proposition 9] for the UIPQ.…”

Section: :: Recurrence Of Random Planar Mapsmentioning

confidence: 99%

Planar Maps, Random Walks and Circle Packing

Nachmias

2020

Lecture Notes in Mathematics

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“…Therefore I is lower semi-continuous. It remains to verify (5). Fix an arbitrary closed (and therefore compact) A ⊂ S, and let δ > 0.…”

Section: Large Deviationsmentioning

confidence: 99%

“…As such this sequence of random variables has a corresponding sequence P n of probability measures and one can ask whether P n satisfies the Large Deviations principle according to the definition given in (5). If this is the case the sequence of graphs G n is defined to be LD-convergent.…”

Section: Definition and Basic Propertiesmentioning

confidence: 99%

“…For sparse graphs, different notions of graph similarity (and corresponding notions of convergence) are of interest in different contexts. In probability theory, the local structure of a graph is often of interest, which suggested the notion of local neighborhood or Benjamini-Schramm convergence, something which has been the subject of much recent study [2][3][4][5]7]. It is not hard to see that, for bounded degree graphs, this notion is equivalent to an a priori different notion of sparse graph convergence, the notion of left-convergence introduced in [12], which takes two graphs to be similar if they have similar subgraph counts, and hence is also natural in the context of combinatorics.…”

Section: Introductionmentioning

confidence: 99%

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Convergent sequences of sparse graphs: A large deviations approach

Borgs

Chayes

Gamarnik³

2016

Random Struct. Alg.

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Models based on sparse graphs are of interest to many communities: they appear as basic models in combinatorics, probability theory, optimization, statistical physics, information theory, and more applied fields of social sciences and economics. Different notions of similarity (and hence convergence) of sparse graphs are of interest in different communities. In probability theory and combinatorics, the notion of Benjamini-Schramm convergence, also known as left-convergence, is used quite frequently. Statistical physicists are interested in the the existence of the thermodynamic limit of free energies, which leads naturally to the notion of right-convergence. Combinatorial optimization problems naturally lead to so-called partition convergence, which relates to the convergence of optimal values of a variety of constraint satisfaction problems. The relationship between these different notions of similarity and convergence is, however, poorly understood.In this paper we introduce a new notion of convergence of sparse graphs, which we call Large Deviations or LD-convergence, and which is based on the theory of large deviations. The notion is introduced by "decorating" the nodes of the graph with random uniform i.i.d. weights and constructing corresponding random measures on [0, 1] and [0, 1] 2 . A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on [0, 1] d , d = 1, 2. The corresponding large deviations rate function can be interpreted as the limit object of the sparse graph sequence. In particular, we can express the limiting free energies in terms of this limit object. We then establish that LD-convergence implies the other three notions of convergence discussed above, and at the same time establish several previously unknown relationships between the other notions of convergence. In particular, we show that partition-convergence does not imply left-or right-convergence, and that right-convergence does not imply partition-convergence.

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Uniform Infinite Planar Triangulations

Abstract: The existence of the weak limit as n → ∞ of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random triangulations of the plane.

Cited by 174 publications

References 32 publications

The continuum random tree is the scaling limit of unlabeled unrooted trees

The continuum random tree is the scaling limit of unlabeled unrooted trees

Planar Maps, Random Walks and Circle Packing

Convergent sequences of sparse graphs: A large deviations approach

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