Recent progress in coalescent theory

Berestycki, Nathanaël

doi:10.21711/217504322009/em161

Cited by 135 publications

(158 citation statements)

References 133 publications

(242 reference statements)

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“…, n}) using a so-called Aldous' construction. We present its simplified version here in a form borrowed from [7], see Section 2.1.3 therein. Start with a chain of length n connecting the labeled vertices U 1 ,U 2 , .…”

Section: Random Compositions and Recordsmentioning

confidence: 99%

“…, n} an edge picked uniformly at random among existing n − k + 1 edges, results in a consistent (in k) family of random compositions given by the sizes of connected components counted from left to right. The number of blocks at time k (or in the k-th row) is k. According to Lemma 2.1 in [7], a composition obtained after removing k − 1 edges is uniformly distributed on the set of all partitions of n into k summands. Note that this construction is also consistent in n in the following sense.…”

Section: Random Compositions and Recordsmentioning

confidence: 99%

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Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Kabluchko¹,

Marynych²

2021

Preprint

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Let ξ 1 , ξ 2 , . . . be a sequence of independent copies of a random vector in R d having an absolutely continuous distribution. Consider a random walk S i := ξ 1 + • • • + ξ i , and let C n,d := conv(0, S 1 , S 2 , . . . , S n ) be the convex hull of the first n + 1 points it has visited. The polytope C n,d is called k-neighborly if for every indices 0We study the probability that C n,d is k-neighborly in various high-dimensional asymptotic regimes, i.e. when n, d, and possibly also k diverge to ∞. There is an explicit formula for the expected number of k-dimensional faces of C n,d which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of C n,d . This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.

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Section: Random Compositions and Recordsmentioning

confidence: 99%

Section: Random Compositions and Recordsmentioning

confidence: 99%

Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Kabluchko¹,

Marynych²

2021

Preprint

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“…That is, i, j ∈ {1, .., m} are in the same block of π κ (t) if and only if U i κ and U j κ share an ancestor at time t. The coalescent tree is the partition-valued process (π κ (t)) t≥0 . Representing coalescent trees as partition-valued processes is standard [5].…”

Section: Coalescent Treementioning

confidence: 99%

“…The class of population models giving rise to Kingman's coalescent is nowhere near as wide as the world of biological populations. This is one reason why coalescent theory has continued to grow with exploration of a rich collection of population models and their coalescent trees [3,5]. Branching processes form a class of population models whose coalescent trees have been extensively studied [6,7,8,9,10,11,12,13,14].…”

Section: Introductionmentioning

confidence: 99%

The coalescent tree of a Markov branching process with generalised logistic growth

Cheek¹

2020

Preprint

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We consider a class of density-dependent branching processes which generalises exponential, logistic and Gompertz growth. A population begins with a single individual, grows exponentially initially, and then growth may slow down as the population size moves towards a carrying capacity. A fixed number of individuals are sampled at a time while the population is still growing superlinearly. Taking the sampling time and carrying capacity simultaneously to infinity, we prove convergence of the coalescent tree. The limiting coalescent tree is in a sense universal over our class of models.

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“…The Bolthausen-Sznitman coalescent[40] is a process on the partitions of N, in which a proportion p of the blocks merge to a single one at rate p ´2 dp. See[26] and[24] for an introduction to coalescent processes.…”

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confidence: 99%

Branching Brownian motion with selection

Maillard¹

2012

Preprint

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Je tiens à remercier chaleureusement mes deux rapporteurs Andreas Kyprianou et Ofer Zeitouni ainsi que les examinateurs Brigitte Chauvin, Francis Comets, Bernard Derrida et Yueyun Hu. J'admire les travaux de chacun et c'est un grand honneur qu'ils aient accepté d'évaluer mon travail.Merci à mon directeur Zhan Shi qui m'a fait découvrir un sujet aussi riche et passionnant que le mouvement brownien branchant, qui m'a toujours soutenu et prodigué ses conseils, tout en m'accordant une grande liberté dans mes recherches. Je remercie également les autres professeurs de l'année 2008/2009 du Master 2 Processus Stochastiques à l'UPMC pour la formation exceptionnelle en probabilités sans laquelle cette thèse n'aurait pas été possible.Certaines autres personnes ont eu une influence directe sur mes recherches mathématiques. Je remercie vivement Julien Berestycki qui, par son enthousiasme infini, a su donner une impulsion nouvelle à mes recherches. Merci à Olivier Hénard, mon ami, pour avoir partagé à de nombreuses occasions sa fine compréhension des processus de branchement.

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Recent progress in coalescent theory

Cited by 135 publications

References 133 publications

Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

The coalescent tree of a Markov branching process with generalised logistic growth

Branching Brownian motion with selection

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