On properties of random dissections and triangulations

Bernasconi, Nicla; Παναγιώτου, Κωνσταντίνος; Steger, Angelika

doi:10.1007/s00493-010-2464-8

Cited by 35 publications

(64 citation statements)

References 9 publications

(15 reference statements)

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“…Probabilistic aspects of uniformly distributed random triangulations have been investigated; see, for example, the articles [18,19] which study graphtheoretical properties of uniform triangulations (such as the maximal vertex degree or the number of vertices of degree k). Graph-theoretical properties of uniform dissections of P n have also been studied, extending the previously mentioned results for triangulations (see [3,10]). From a more geometrical point of view, Aldous [1,2] studied the shape of a large uniform triangulation viewed as a random compact subset of the closed unit disk.…”

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confidence: 58%

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Random stable laminations of the disk

Kortchemski¹

2014

Ann. Probab.

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We study large random dissections of polygons. We consider random dissections of a regular polygon with n sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index θ ∈ (1, 2]. As n goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If θ = 2, we recover Aldous' Brownian triangulation. However, if θ ∈ (1, 2), large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive Lévy process of index θ. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely 2 − 1/θ.Introduction. In this article we study large random dissections of polygons. A dissection of a polygon is the union of the sides of the polygon and of a collection of diagonals that may intersect only at their endpoints. The faces are the connected components of the complement of the dissection in the polygon. The particular case of triangulations (when all faces are triangles) has been extensively studied in the literature. For every integer n ≥ 3, let P n be the regular polygon with n sides whose vertices are the nth roots of unity. It is well known that the number of triangulations of P n is the Catalan number of order n − 2. In the general case, where faces of degree greater than three are allowed, there is no known explicit formula for the number of dissections of P n , although an asymptotic estimate is known (see [10,17]). Probabilistic aspects of uniformly distributed random triangulations have been investigated; see, for example, the articles [18,19] which study graphtheoretical properties of uniform triangulations (such as the maximal vertex degree or the number of vertices of degree k). Graph-theoretical properties

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confidence: 58%

“…Put µ 1 = 0 and µ 0 = 1 − ∞ j=2 µ j so that µ = (µ j ) j≥0 defines a probability measure on N, which satisfies the assumptions of Definition 1.3. Let n ≥ 2 and let Z n be defined as in (3). Then Z n > 0 if, and only if, P µ [λ(τ ) = n] > 0.…”

Section: Random Dissections and Galton-watson Treesmentioning

confidence: 99%

Random stable laminations of the disk

Kortchemski¹

2014

Ann. Probab.

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“…By traversing the edges of the root face in clock-wise order, any edge-rooted dissection of a polygon may be decomposed into an ordered sequence of C out -objects. This decomposition was previously established in [BPS10]. In order for the sizes to add up correctly, we require the root vertex to be replaced by a * -vertex that does not contribute to the total number of vertices.…”

Section: Enumeration Constantsmentioning

confidence: 99%

Scaling limits of random outerplanar maps with independent link-weights

Stufler¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. The scaling limit of large simple outerplanar maps was established by Caraceni using a bijection due to Bonichon, Gavoille and Hanusse. The present paper introduces a new bijection between outerplanar maps and trees decorated with ordered sequences of edgerooted dissections of polygons. We apply this decomposition in order to provide a new, short proof of the scaling limit that also applies to the general setting of first-passage percolation. We obtain sharp tail-bounds for the diameter and recover the asymptotic enumeration formula for outerplanar maps. Our methods also enable us to treat subclasses such as bipartite outerplanar maps.

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“…One substantial benefit of such a decomposition is that it enables us to develop mechanically algorithms that sample maps from the family in question by using the framework of Boltzmann samplers. Such sampling algorithms are an important ingredient in our proofs, and were used for the first time systematically by Bernasconi, the second author, and Steger in [3] to study properties of random structures. The Boltzmann model was introduced by Duchon, Flajolet, Louchard and Schaeffer in [6].…”

Section: Random Maps In the Boltzmann Modelmentioning

confidence: 99%

Vertices of Degree k in Random Maps

Johannsen¹,

Παναγιώτου²

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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This work is devoted to the study of the typical structure of a random map. Maps are planar graphs embedded in the plane. We investigate the degree sequences of random maps from families of a certain type, which, among others, includes fundamental map classes like those of biconnected maps, 3-connected maps, and triangulations. In particular, we develop a general framework that allows us to derive relations and exact asymptotic expressions for the expected number of vertices of degree k in random maps from these classes, and also provide accompanying large deviation statements. Extending the work of Gao and Wormald (Combinatorica, 2003) on random general maps, we obtain as results of our framework precise information about the number of vertices of degree k in random biconnected, 3-connected, loopless, and bridgeless maps.

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On properties of random dissections and triangulations

Cited by 35 publications

References 9 publications

Random stable laminations of the disk

Random stable laminations of the disk

Scaling limits of random outerplanar maps with independent link-weights

Vertices of Degree k in Random Maps

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