Results 1  10
of
70
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
Abstract

Cited by 48 (3 self)
 Add to MetaCart
Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
On the Structure of QuasiStationary Competing Particles Systems
, 2007
"... We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {qij}i,j∈N. A probability measure on th ..."
Abstract

Cited by 41 (4 self)
 Add to MetaCart
(Show Context)
We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {qij}i,j∈N. A probability measure on the pair (X, Q) is said to be quasistationary if the joint law of the gaps of X and of Q is invariant under the evolution. A known class of universally quasistationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchally nested PoissonDirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasistationary. The main result of this work is a proof of this conjecture for the case where qij assume only a finite number of values. The result is of relevance for meanfield spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchal organization of the Gibbs measure was first proposed as an ansatz.
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models
 Ann. Probab
, 2008
"... Given any regularly varying dislocation measure, we identify a natural selfsimilar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s betasplitting models and Ford’s alpha models for phyl ..."
Abstract

Cited by 40 (13 self)
 Add to MetaCart
(Show Context)
Given any regularly varying dislocation measure, we identify a natural selfsimilar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s betasplitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.
The structure of the allelic partition of the total population for GaltonWatson processes with neutral mutations
"... We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
(Show Context)
We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and the number of mutantchildren of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given. 1. Introduction. We consider a Galton–Watson process, that is, a population model with asexual reproduction such that at every generation, each individual gives birth to a random number of children according to a fixed distribution and independently of the other individuals in the population. We are interested in the situation where a child can be either a clone, that
SPINAL PARTITIONS AND INVARIANCE UNDER REROOTING OF CONTINUUM RANDOM TREES
, 2009
"... We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees ..."
Abstract

Cited by 25 (12 self)
 Add to MetaCart
(Show Context)
We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform rerooting.
Regenerative partition structures
, 2008
"... We consider Kingman’s partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by sizebiased sampling. We a ..."
Abstract

Cited by 16 (7 self)
 Add to MetaCart
We consider Kingman’s partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by sizebiased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the twoparameter family of partition structures.
Fluctuation theory and exit systems for positive selfsimilar Markov processes
 Preprint. AND ASYMPTOTIC nTUPLE LAWS AT FIRST AND LAST PASSAGE 563
, 2009
"... For a positive selfsimilar Markov process, X, we construct a local time for the random set, �, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
For a positive selfsimilar Markov process, X, we construct a local time for the random set, �, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) associated to a positive selfsimilar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set � and the process X sampled on the local time scale. The process (R, H) is described in terms of a ladder process linked to the Lévy process associated to X via Lamperti’s transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finitedimensional convergence of (R, H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012–1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Lévy process oscillates. 1. Introduction. In
Limit distributions for large Pólya urns
, 2011
"... We consider a twocolor Pólya urn in the case when a fixed number S of balls is added at each step. Assume it is a large urn that is, the second eigenvalue m of the replacement matrix satisfies 1/2 <m/S ≤ 1. After n drawings, the composition vector has asymptotically a first deterministic term of ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
(Show Context)
We consider a twocolor Pólya urn in the case when a fixed number S of balls is added at each step. Assume it is a large urn that is, the second eigenvalue m of the replacement matrix satisfies 1/2 <m/S ≤ 1. After n drawings, the composition vector has asymptotically a first deterministic term of order n and a second random term of order n m/S. The object of interest is the limit distribution of this random term. The method consists in embedding the discretetime urn in continuous time, getting a twotype branching process. The dislocation equations associated with this process lead to a system of two differential equations satisfied by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitly related to Abelian integrals over the Fermat curve of degree m. The limit laws appear to constitute a new family of probability densities supported by the whole real line.
Random recursive triangulations of the disk via fragmentation theory
 ANNALS OF PROBABILITY
, 2011
"... We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing ..."
Abstract

Cited by 14 (12 self)
 Add to MetaCart
(Show Context)
We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension β ∗ + 1, where β ∗ = ( √ 17 − 3)/2, and that it can be described as the geodesic lamination coded by a random continuous function which is Hölder continuous with exponent β ∗ − ε, for every ε> 0. We also discuss recursive constructions of triangulations of the ngon that give rise to the same continuous limit when n tends to infinity.