Random gluing of metric spaces

Sénizergues, Delphin

doi:10.1214/19-aop1348

Cited by 11 publications

(15 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…provided that k is large enough, since |B k | = n 1−α+•(1) k = n •(1) k+1 , by (14) and (12). On the other hand, again by Corollary 22,…”

Section: Proof Of Proposition 17mentioning

confidence: 72%

See 1 more Smart Citation

Random trees constructed by aggregation

Curien¹,

Haas²

2017

Ann. inst. Fourier

View full text Add to dashboard Cite

We study a general procedure that builds random R-trees by gluing recursively a new branch on a uniform point of the pre-existing tree. The aim of this paper is to see how the asymptotic behavior of the sequence of lengths of branches influences some geometric properties of the limiting tree, such as compactness and Hausdorff dimension. In particular, when the sequence of lengths of branches behaves roughly like n −α for some α ∈ (0, 1], we show that the limiting tree is a compact random tree of Hausdorff dimension α −1 . This encompasses the famous construction of the Brownian tree of Aldous. When α > 1, the limiting tree is thinner and its Hausdorff dimension is always 1. In that case, we show that α −1 corresponds to the dimension of the set of leaves of the tree. RésuméNous nous intéressons à une procédure générale de construction d'arbres réels aléatoires par collages successifs de nouvelles branches. A chaque étape, la nouvelle branche est collée en un point choisi uniformément sur l'arbre pré-existant. Notre objectif principal est de comprendre comment le comportement asymptotique de la suite des longueurs de branches influence certaines propriétés géométriques de l'arbre, telles que la compacité ou la dimension de Hausdorff. Nous montrons en particulier que lorsque la suite de longueurs de branches se comporte en n −α , avec α ∈ (0, 1] fixé, l'arbre limite est compact, de dimension de Hausdorff α −1 . A titre d'exemple, ceci englobe une construction bien connue de l'arbre brownien d'Aldous. Lorsque α > 1, l'arbre limite est plus fin et de dimension de Hausdorff 1. Dans ce cas, nous montrons que α −1 correspond à la dimension de l'ensemble des feuilles de l'arbre.

show abstract

“…provided that k is large enough, since |B k | = n 1−α+•(1) k = n •(1) k+1 , by (14) and (12). On the other hand, again by Corollary 22,…”

Section: Proof Of Proposition 17mentioning

confidence: 72%

“…Corollary 22. Assume that the sequence (n k ) satisfies n k+1 ≥ g(n k ) for all k -where g is the function of the previous lemma -as well as (12). For each k ∈ N, each r > 0 and each x ∈ B k , consider the total length of branches of B k+1 that are grafted on B(x, r) ∩ B k ⊂ T n k+1 −1 .…”

Section: Lengths Estimatesmentioning

confidence: 99%

Random trees constructed by aggregation

Curien¹,

Haas²

2017

Ann. inst. Fourier

View full text Add to dashboard Cite

show abstract

“…which indeed coincides with the above limit of E exp λ ξ n (t) − ξ n (s) . P |ξ n ((τ + θ) ∧ 1) − ξ n (τ )| > ε = 0 (14) where S n is the set of stopping times with respect to the filtration generated by the process ξ n .…”

Section: Functional Convergencementioning

confidence: 99%

Asymptotics of heights in random trees constructed by aggregation

Haas¹

2017

Electron. J. Probab.

View full text Add to dashboard Cite

To each sequence (a n ) of positive real numbers we associate a growing sequence (T n ) of continuous trees built recursively by gluing at step n a segment of length a n on a uniform point of the pre-existing tree, starting from a segment T 1 of length a 1 . Previous works [4,8] on that model focus on the influence of (a n ) on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence (a n ) is regularly varying with a non-negative index, so that the sequence (T n ) exploses. We determine the asymptotics of the height of T n and of the subtrees of T n spanned by the root and points picked uniformly at random and independently in T n , for all ∈ N.

show abstract

“…This is done by Sénizergues [33] who gives a sufficient condition for recursive constructions of graphs to converge almost surely in the scaling limit. We emphasize that his result includes our setting here, hence proving the a.s. scaling limit of the trees T n , but that it does not identify the limit as a multi-type fragmentation tree, but rather as a gluing of random metric spaces as studied in [34]. His sufficient condition can be stated as follows in our setting: for N i , i ≥ 1 a sequence of i.i.d.…”

Section: Application 1: Growing Models Of Random Treesmentioning

confidence: 99%

Scaling Limits of Markov-Branching Trees and Applications

Haas¹

2018

XII Symposium of Probability and Stochastic Processes

View full text Add to dashboard Cite

We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and their type and give rise to (size,type)-children in a Galton-Watson fashion, with the rule that the size of any individual is a least the sum of the sizes of its children. Assuming that the macroscopic size-splittings are rare, we describe the scaling limits of multi-type Markov Branching trees in terms of multi-type self-similar fragmentation trees. We observe three different regimes according to whether the probability of type change of a size-biased child is proportional to the probability of macroscopic splitting (the critical regime, in which we get in the limit multi-type fragmentation trees with indeed several types), smaller than the probability of macroscopic splitting (the solo regime, in which the limit trees are monotype as we never see a type change), or larger than the probability of macroscopic splitting (the mixing regime, in which case the types mix in the limit and we get monotype fragmentation trees).This framework allows us to unify models which may a priori seem quite different, a strength which we illustrate with two notable applications. The first one concerns the description of the scaling limits of growing models of random trees built by gluing at each step on the current structure a finite tree picked randomly in a finite alphabet of trees, extending Rémy's wellknown algorithm for the generation of uniform binary trees to a fairly broad framework. We are then either in the critical regime with multi-type fragmentation trees in the scaling limit, or in the solo regime. The second application concerns the scaling limits of large multi-type critical Galton-Watson trees when the offspring distributions all have finite second moments. This topic has already been studied but our approach gives a different proof and we improve on previous results by relaxing some hypotheses. We are then in the mixing regime: the scaling limits are always multiple of the Brownian CRT, a pure monotype fragmentation tree in our framework.

show abstract

Random gluing of metric spaces

Cited by 11 publications

References 21 publications

Random trees constructed by aggregation

Random trees constructed by aggregation

Asymptotics of heights in random trees constructed by aggregation

Scaling Limits of Markov-Branching Trees and Applications

Contact Info

Product

Resources

About