Percolation on random triangulations and stable looptrees

Curien, Nicolas; Kortchemski, Igor

doi:10.1007/s00440-014-0593-5

Cited by 47 publications

(93 citation statements)

References 36 publications

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“…is smaller than n 3 oe(n) = oe(n) by (11). On the other hand, conditionally on λ(τ • ) = n, the probability that there exists a couple of leaves of τ • which does not belong to (L j , L j ) 1≤j≤n 3 is smaller than n 2 1 − n −2 n 3 = oe(n).…”

Section: Proof Of Theoremmentioning

confidence: 91%

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The CRT is the scaling limit of random dissections

Curien

Haas

Kortchemski

2014

Random Struct. Alg.

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We study the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissections or uniform p-angulations. As their number of vertices n goes to infinity, we show that these random graphs, rescaled by n −1/2 , converge in the Gromov-Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. The scaling constant depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Markov chain which compares the length of geodesics in dissections with the length of geodesics in their dual trees.

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Section: Proof Of Theoremmentioning

confidence: 91%

“…We leave details to the reader. The motivation of this model comes from the fact that Theorem 14 has an interesting application to the study of the asymptotic behavior of subcritical sitepercolation on large random triangulations [11].…”

Section: Extensions and Discrete Looptreesmentioning

confidence: 99%

The CRT is the scaling limit of random dissections

Curien

Haas

Kortchemski

2014

Random Struct. Alg.

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“…Remark 3.5 (Links with [11]). The above proposition in the case y ≤ṗ c could directly be deduced from [11] and probably also for y >ṗ c with a little more work. Notice in particular that the weights [x n ]T(x, y) can be related to Q a ({triangulations with a boundary of length n}) where the measure Q a is defined in [11,Eq.…”

Section: Triangulations With Boundary and Outer Verticesmentioning

confidence: 99%

“…This approach was first developed in the pioneer work of Angel [1] for site-percolation on the Uniform Infinite Planar Triangulation (UIPT) and later extended to other models of percolation and maps [2,10,20,24]. As opposed to the "dynamical" approach of the peeling process, the work [11] uses a "fixed" combinatorial decomposition (inspired by [5]) and known enumeration results on triangulations to study the scaling limit of percolation cluster conditioned on having a large boundary. All the above works focused, in a sense, on the geometry of one percolation interface, hence studied the geometry of the outer boundary of a large percolation cluster.…”

Section: Introductionmentioning

confidence: 99%

A Boltzmann Approach to Percolation on Random Triangulations

Bernardi¹,

Curien²,

Miermont

2019

Can. J. Math.-J. Can. Math.

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We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bondpercolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length n decays exponentially with n except at a particular value pc of the percolation parameter p for which the decay is polynomial (of order n −10/3 ). Moreover, the probability that the origin cluster has size n decays exponentially if p < pc and polynomially if p ≥ pc.The critical percolation value is pc = 1/2 for site percolation, and pc = 2

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“…In the recent years, it has been realized that BGW trees in which a condensation phenomenon occurs code a variety of random combinatorial structures such as random planar maps [AB15,JS15,Ric18], outerplanar maps [SS17], supercritical percolation clusters of random triangulations [CK15] or minimal factorizations [FK17]. See [Stu16] for a combinatorial framework and further examples.…”

Section: Contextmentioning

confidence: 99%

Condensation in critical Cauchy Bienaymé–Galton–Watson trees

Kortchemski¹,

Richier²

2019

Ann. Appl. Probab.

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We are interested in the structure of large Bienaymé-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index α = 1. In stark contrast to the case α ∈ (1, 2], we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges. To this end, we establish limit theorems for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. These results are of independent interest. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called non-generic of parameter 3/2). This supports the conjecture that faces in Le Gall & Miermont's 3/2-stable maps are self-avoiding.2010 Mathematics Subject Classification. -Primary 60J80 · 60G50 · 60F17 · 05C05; Secondary 05C80 · 60C05.

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Percolation on random triangulations and stable looptrees

Cited by 47 publications

References 36 publications

The CRT is the scaling limit of random dissections

The CRT is the scaling limit of random dissections

A Boltzmann Approach to Percolation on Random Triangulations

Condensation in critical Cauchy Bienaymé–Galton–Watson trees

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