The scaling limit of random simple triangulations and random simple quadrangulations

Addario‐Berry, Louigi; Albenque, Marie

doi:10.1214/16-aop1124

Cited by 59 publications

(165 citation statements)

References 30 publications

(81 reference statements)

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“…Let us mention that other convergences toward the Brownian map similar to Theorem 1 have been obtained using also other bijections with labeled trees: Beltran and Le Gall [8] studied random quadrangulations without vertices of degree one, Addario-Berry and Albenque [3] considered random triangulations and quadrangulations without loops or multiple edges and Bettinelli, Jacob, and Miermont [10] uniform random maps with n edges.…”

Section: Approach and Organization Of The Papermentioning

confidence: 84%

“…Observe that LR(m) = 1 + i≥1 (i − 1)m i = 2 + (r − 2) + i≥1 (i − 1)m i denotes the number of trees in the forest obtained from T n by removing the reduced tree T n (u n , v n ) when A(u n , v n ) = (m (1) , m (2) , m (3) ) and k wn = r: there are i − 1 components for each of the m i elements of ∅, w n ∪ û n , u n ∪ v n , v n with i children, as well as r − 2 components corresponding to the children of w n different fromû n andv n , and the two components above u n and v n . As previously, the triplet (T n , u n , v n ) is characterized by the forest obtained by removing the reduced tree T n (u n , v n ) and the content of the latter, which is Cont(u n , v n ) plus the information (k wn , χû n , χv n ) about the branch-point.…”

Section: Proof Of Lemmamentioning

confidence: 99%

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Scaling limits of random bipartite planar maps with a prescribed degree sequence

Marzouk

2018

Random Struct Algorithms

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We study the asymptotic behavior of uniform random maps with a prescribed face-degree sequence, in the bipartite case, as the number of faces tends to infinity. Under mild assumptions, we show that, properly rescaled, such maps converge in distribution toward the Brownian map in the Gromov-Hausdorff sense. This result encompasses a previous one of Le Gall for uniform random q-angulations where q is an even integer. It applies also to random maps sampled from a Boltzmann distribution, under a second moment assumption only, conditioned to be large in either of the sense of the number of edges, vertices, or faces. The proof relies on the convergence of so-called "discrete snakes" obtained by adding spatial positions to the nodes of uniform random plane trees with a prescribed child sequence recently studied by Broutin and Marckert. This paper can alternatively be seen as a contribution to the study of the geometry of such trees. K E Y W O R D SBrownian map; Brownian snake; labelled trees; limit theorems; random maps Random Struct Alg. 2018;00:1-56.wileyonlinelibrary.com/journal/rsadefines a probability measure on M. We consider next such random maps conditioned to have a large size for several notions of size. For every integer n ≥ 1, let M E=n , M V =n and M F=n be the subsets of M of those maps with respectively n edges, n vertices and n faces. For every S = {E, V , F} and every n ≥ 1, we definethe law of a Boltzmann map conditioned to have size n; here and later, we shall always, if necessary, implicitly restrict ourselves to those values of n for which W q (M S=n ) � = 0, and limits shall be understood along this subsequence. Under mild integrability conditions on q, we prove in Section 7 that for every S ∈ {E, V , F}, there exists a constant K q S > 0 such that if M n is sampled from P q S=n for every n ≥ 1, then the convergence in distributionholds in the sense of Gromov-Hausdorff. We refer to Theorem 3 for a precise statement. Observe that for any choice S ∈ {E, V , F}, if M n is sampled from P q S=n then, conditional on its degree sequence, say, ν Mn = (ν Mn (i); i ≥ 1), it has the uniform distribution in M(ν Mn ). The proof of the above convergence consists in showing that ν Mn satisfies (H) in probability for some deterministic limit law p q . Indeed, by Skorohod's representation Theorem, there exists then a probability space where versions of ν Mn under P q S=n satisfy (H) almost surely so we may apply Theorem 1 and conclude the convergence in law of the rescaled maps.The case S = V was obtained by Le Gall [27, Theorem 9.1], relying on results of Marckert and Miermont [33], when q is regular critical, meaning that the distribution p q (which is roughly that of the half-degree of a typical face when we see vertices as faces of degree 0) admits small exponential moments. Here, we generalize this result (and consider other conditionings) to all generic critical sequences q, that is, those for which p q admits a second moment.

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Section: Approach and Organization Of The Papermentioning

confidence: 84%

Section: Proof Of Lemmamentioning

confidence: 99%

Scaling limits of random bipartite planar maps with a prescribed degree sequence

Marzouk

2018

Random Struct Algorithms

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“…(In [13,21], the convergence is only stated for the GH topology, but the proof in fact yields the above formulation. This is also stated explicitly in [3,Theorem 4.1].) So, for the pointed GHP topology,…”

Section: Appendix B Convergence To the Brownian Plane For The Local mentioning

confidence: 78%

“…We thus need the notion of local GHP topology, described below. See [3,11,17,20,21] for details on the GH(P) topology, […”

Section: Convergence In the Local Ghp Topologymentioning

confidence: 99%

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The Brownian plane with minimal neck baby universe

Wen

2017

Random Struct Algorithms

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For each n ∈ N, let Q n be a uniform rooted quadrangulation, endowed with an appropriate measure, of size n conditioned to have r(n) vertices in its root block. We prove that for a suitable function r(n), after rescaling graph distance by 21 40·r(n) 1/4 , Q n converges to a random pointed noncompact metric measure space S, in the local Gromov-Hausdorff-Prokhorov topology. The space S is built by identifying a uniform point of the Brownian map with the distinguished point of the Brownian plane.

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A Taste of Scaling Limit

Curien

2023

Lecture Notes in Mathematics

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The scaling limit of random simple triangulations and random simple quadrangulations

Cited by 59 publications

References 30 publications

Scaling limits of random bipartite planar maps with a prescribed degree sequence

Scaling limits of random bipartite planar maps with a prescribed degree sequence

The Brownian plane with minimal neck baby universe

A Taste of Scaling Limit

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