Periodic Pólya urns, the density method and asymptotics of Young tableaux

Banderier, Cyril; Marchal, Philippe; Wallner, Michael

doi:10.1214/19-aop1411

Cited by 2 publications

(21 citation statements)

References 75 publications

(193 reference statements)

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“…For some regular sequences of fitnesses, we can explicitly describe the distribution of the limiting sequence using Beta, Gamma and Mittag-Leffler distributions. This relates in various ways to other results that can be found in the literature associated to preferential attachment trees or to urn models, contained in [36,27,41,26,42,40,39,1].…”

Section: Introductionsupporting

confidence: 69%

“…The distribution of the limiting sequence (m a n ) n≥1 can be characterized, and even has a reasonable description for certain regular sequences of fitnesses a, as it is explained in the following paragraph. This result is actually related to the study of some urn models like the Pólya urns with immigration of [42] or the periodic Pólya urns of [1] and allows us to provide some alternative proofs and complete some of the known results about those processes. This is developed in Section 5.2.…”

Section: Geometry Of Wrts and Patsmentioning

confidence: 91%

“…Furthermore, if a n ≤ (n + 1) c +o (1) , for some 0 ≤ c < 1 c+1 , the previous convergence also takes place in the space p of p-th power summable sequences, for all p > c+1 1−(c+1)c .…”

Section: Geometry Of Wrts and Patsmentioning

confidence: 99%

“…Definitions. For any sequence of non-negative real numbers (w n ) n≥1 with w 1 > 0, we define the distribution WRT((w n ) n≥1 ) on sequences of growing rooted labelled trees 1 , which is called the weighted recursive tree with weights (w n ) n≥1 . We construct a sequence of rooted trees (T n ) n≥1 by starting from T 1 containing only one root-vertex u 1 with label 1 and letting it evolve in the following manner: the tree T n+1 is obtained from T n by adding a vertex u n+1 with label n + 1.…”

Section: Two Related Models Of Growing Treesmentioning

confidence: 99%

“…We assume that we work with a sequence w that satisfies the following assumption ( p γ ) for some γ > 0 and p ∈ (1 , 2], (1) .…”

Section: Height and Profile Of Wrtmentioning

confidence: 99%

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Geometry of weighted recursive and affine preferential attachment trees

Sénizergues

2021

Electron. J. Probab.

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We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n ≥ 2 a new vertex un is added to the tree and its parent is chosen randomly according to some rule. In the weighted recursive tree, we choose the parent u k of un among {u1, u2, . . . , un−1} with probability proportional to w k , where (wn) n≥1 is some deterministic sequence that we fix beforehand.In the affine preferential attachment tree with fitnesses, the probability of choosing any u k is proportional to a k + deg + (u k ), where deg + (u k ) denotes its current number of children, and the sequence of fitnesses (an) n≥1 is deterministic and chosen as a parameter of the model. We show that for any sequence (an) n≥1 , the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a random sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.

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