Simply Generated Non-Crossing Partitions

Kortchemski, Igor; Marzouk, Cyril

doi:10.1017/s0963548317000050

Cited by 14 publications

(16 citation statements)

References 39 publications

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“…, N} and consider the j-th white corner of T : it is a sector around a white vertex delimited by two consecutive edges, whose other extremity is therefore black; consider the previous black corner in contour order, in the construction of the JS bijection, an edge of T starts from this corner and we claim that the other extremity of this edge is u j the j-th vertex of T in lexicographical order. We refer to the proof of Proposition 2.1 and Figure 4 in [23]. It follows that if c • j ∈ •(T ) is the white vertex of T visited at the j-th step in the white contour sequence, then the image of u j by the JS bijection is…”

Section: The Janson-stefánsson Bijectionmentioning

confidence: 98%

“…, N}, we let H(j) denote the number of strict ancestors of u j whose last child is not an ancestor of u j , that is, Proof Let us first prove the equality of the label processes. We use the observation from [23] that the lexicographical order on the vertices of T corresponds to the contour order on the black corners of T which, by a shift, corresponds to the contour order on the white corners of T . Specifically, let N be the number of edges of both trees, fix j ∈ {0, .…”

Section: The Janson-stefánsson Bijectionmentioning

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: The Janson-stefánsson Bijectionmentioning

confidence: 98%

Section: The Janson-stefánsson Bijectionmentioning

confidence: 99%

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

Marzouk

2018

Random Struct Algorithms

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“…Recall that Motzkin paths are lattice paths in the plane starting from the origin up to (n, 0), using up steps (1, 1), down steps (1, −1), and horizontal steps (1, 0), and never going below the x-axis. Identifying k ∈ [n] with e −2ikπ/n , then non-crossing partitions can also be seen as compact subsets of the unit disk [43].…”

Section: Partitions and Bell Polynomialsmentioning

confidence: 99%

An Introduction to Wishart Matrix Moments

Bishop

Moral

Niclas

2018

FNT in Machine Learning

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“…Let us now turn to the proof of (17). First, we observe that for 0 < ε < a −1 1 , we have that P (1) alt #T n − 1 a 1 > ε, #T (1) = n = P (1) alt #T > 1 a 1 + ε n, #T (1) = n + P (1) alt #T < 1 a 1 − ε n, #T (1) = n .…”

Section: Alternating Two-type Gw Treementioning

confidence: 92%

“…where H exc is the normalized excursion of the continuous-time height process process associated with a strictly stable spectrally positive Lévy process Y In recent years, this special family of two-type GW trees has been the subject of many studies due to their remarkable relationship with the study of several important objects and models of growing relevance in modern probability such that random planar maps [20], percolation on random maps [7], non-crossing partitions [17], to mention just a few. On the other hand, up to our knowledge the result of Theorem 3 has not been proved before under our assumptions on the offspring distribution.…”

Section: Alternating Two-type Gw Treementioning

confidence: 99%

On scaling limits of multitype Galton-Watson trees with possibly infinite variance

Berzunza¹

2018

ALEA

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Abstract. In this work, we study asymptotics of multitype Galton-Watson forests with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive Lévy process. This gives an analog of a result obtained by Miermont (2008) in the case of multitype Galton-Watson forests with finite covariance matrices of the offspring distribution. Our approach relies on a remarkable decomposition for multitype trees into monotype trees introduced in Miermont (2008).

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Simply Generated Non-Crossing Partitions

Cited by 14 publications

References 39 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

An Introduction to Wishart Matrix Moments

On scaling limits of multitype Galton-Watson trees with possibly infinite variance

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