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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 ..."
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Cited by 17 (3 self)
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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
Distributional limits for critical random graphs
 In preparation
, 2009
"... We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metri ..."
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Cited by 12 (5 self)
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We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean. Keywords: Random graphs, GromovHausdorff distance, scaling limits, continuum random tree, diameter. 2000 Mathematics subject classification: 05C80, 60C05.
SIMPLY GENERATED TREES, CONDITIONED GALTON–WATSON TREES, RANDOM ALLOCATIONS AND CONDENSATION (EXTENDED ABSTRACT)
, 2012
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Approximate simulation of Hawkes processes
 In preparation
, 2004
"... This article concerns a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branchi ..."
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Cited by 7 (2 self)
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This article concerns a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structure, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. Examples of applications and empirical results are presented.
Critical behavior in inhomogeneous random graphs
, 2009
"... We study the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities. We show that the critical behavior depends sensitively on the properties of the asymptotic degrees. Indeed, when the proportion of vertices with degree ..."
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Cited by 5 (3 self)
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We study the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities. We show that the critical behavior depends sensitively on the properties of the asymptotic degrees. Indeed, when the proportion of vertices with degree at least k is bounded above by k −τ+1 for some τ> 4, the largest critical connected component is of order n 2/3, where n denotes the size of the graph, as on the ErdősRényi random graph. The restriction τ> 4 corresponds to finite third moment of the degrees. When, the proportion of vertices with degree at least k is asymptotically equal to ck −τ+1 for some τ ∈ (3,4), the largest critical connected component is of order n (τ−2)/(τ−1) , instead. Our results show that, for inhomogeneous random graphs with a powerlaw degree sequence, the critical behavior admits a transition when the third moment of the degrees turns from finite to infinite. Similar phase transitions have been shown to occur for typical distances in such random graphs when the variance of the degrees turns from finite to infinite. We present further results related to the size of the critical or scaling window, and state conjectures for this and related random graph models.
DISTANCES BETWEEN PAIRS OF VERTICES AND VERTICAL PROFILE IN CONDITIONED GALTON–WATSON TREES
"... Abstract. We consider a conditioned Galton–Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularit ..."
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Cited by 5 (1 self)
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Abstract. We consider a conditioned Galton–Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the latter proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet–Mélou and Janson [5], saying that the vertical profile of a randomly labelled conditioned Galton–Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion). 1. Introduction and
Susceptibility in subcritical random graphs
 125207. OF RANDOM GRAPHS WITH GIVEN VERTEX DEGREES 25
"... Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its ..."
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Cited by 5 (4 self)
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Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal. 1.
Random trees with superexponential branching weights
 J. Phys. A: Math. Theor
"... Abstract. We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors wn associated to the vertices of the tree and depending only on their individual degrees n. We focus on the case when wn grows faster than exponentially with n. In this ..."
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Cited by 4 (4 self)
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Abstract. We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors wn associated to the vertices of the tree and depending only on their individual degrees n. We focus on the case when wn grows faster than exponentially with n. In this case the measures on trees of finite size N converge weakly as N tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form wn = ((n − 1)!) α with α> 0 we obtain more refined results about the approach to the infinite volume limit. 1.
Individual displacements for linear probing hashing with different insertion policies
 ACM Transactions on Algorithms
, 2005
"... Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occ ..."
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Cited by 4 (1 self)
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Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occupied cells converging to some α, 0 < α < 1. (In the case of Last Come, the results are more complicated and less complete than in the other cases.) We also show, using the diagonal Poisson transform studied by Poblete, Viola and Munro, that exact expressions for finite m and n can be obtained from the limits as m, n → ∞. We end with some results, conjectures and questions about the shape of the limit distributions. These have some relevance for computer applications. 1.
SUBGAUSSIAN TAIL BOUNDS FOR THE WIDTH AND HEIGHT OF CONDITIONED GALTON–WATSON TREES.
"... Abstract. We study the height and width of a Galton–Watson tree with offspring distribution ξ satisfying E ξ = 1, 0 < Var ξ < ∞, conditioned on having exactly n nodes. Under this conditioning, we derive subGaussian tail bounds for both the width (largest number of nodes in any level) and heig ..."
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Cited by 3 (1 self)
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Abstract. We study the height and width of a Galton–Watson tree with offspring distribution ξ satisfying E ξ = 1, 0 < Var ξ < ∞, conditioned on having exactly n nodes. Under this conditioning, we derive subGaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 ≤ k ≤ n. 1.