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Research schools

We give some results about tree-indexed random walks aka branching random walks. In particular, we investigate the growth of the maximum of such a walk.
Based on joint work with Piotr Dyszewski and Thomas Hofelsauer.

60G50 ; 60J10 ; 60J80

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Research talks;Combinatorics;Mathematical Physics;Probability and Statistics

In the first half of the talk, I will survey results and open problems on transience of self-interacting martingales. In particular, I will describe joint works with S. Popov, P. Sousi, R. Eldan and F. Nazarov on the tradeoff between the ambient dimension and the number of different step distributions needed to obtain a recurrent process. In the second, unrelated, half of the talk, I will present joint work with Tom Hutchcroft, showing that the component structure of the uniform spanning forest in $\mathbb{Z}^d$ changes every dimension for $d > 8$. This sharpens an earlier result of Benjamini, Kesten, Schramm and the speaker (Annals Math 2004), where we established a phase transition every four dimensions. The proofs are based on a the connection to loop-erased random walks. In the first half of the talk, I will survey results and open problems on transience of self-interacting martingales. In particular, I will describe joint works with S. Popov, P. Sousi, R. Eldan and F. Nazarov on the tradeoff between the ambient dimension and the number of different step distributions needed to obtain a recurrent process. In the second, unrelated, half of the talk, I will present joint work with Tom Hutchcroft, showing that the ...

05C05 ; 05C80 ; 60G50 ; 60J10 ; 60K35 ; 82B43

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Research talks;Probability and Statistics

This talk is based on a work jointly with Timothy Budd (Copenhagen), Nicolas Curien (Orsay) and Igor Kortchemski (Ecole Polytechnique).
Consider a self-similar Markov process $X$ on $[0,\infty)$ which converges at infinity a.s. We interpret $X(t)$ as the size of a typical cell at time $t$, and each negative jump as a birth event. More precisely, if ${\Delta}X(s) = -y < 0$, then $s$ is the birth at time of a daughter cell with size $y$ which then evolves independently and according to the same dynamics. In turn, daughter cells give birth to granddaughter cells each time they make a negative jump, and so on.
The genealogical structure of the cell population can be described in terms of a branching random walk, and this gives rise to remarkable martingales. We analyze traces of these mar- tingales in physical time, and point at some applications for self-similar growth-fragmentation processes and for planar random maps.
This talk is based on a work jointly with Timothy Budd (Copenhagen), Nicolas Curien (Orsay) and Igor Kortchemski (Ecole Polytechnique).
Consider a self-similar Markov process $X$ on $[0,\infty)$ which converges at infinity a.s. We interpret $X(t)$ as the size of a typical cell at time $t$, and each negative jump as a birth event. More precisely, if ${\Delta}X(s) = -y < 0$, then $s$ is the birth at time of a daughter cell with size $y$ which then ...

60G51 ; 60G18 ; 60J75 ; 60G44 ; 60G50

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- xi; 221 p.
ISBN 978-1-4704-1945-5

Contemporary mathematics , 0668

Localisation : Collection 1er étage

Philip Feinsilver # Salah-Eldin Mohammed # Arunava Mukherjea # mesure de probabilités # équation différentielle # processus de Markov # géométrie combinatoire

05C50 ; 15A66 ; 54C40 ; 60B15 ; 60G50 ; 60H07 ; 60H15 ; 60H30 ; 60J05

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- xvi; 406 p.
ISBN 978-1-4939-3075-3

Fields institute communications , 0076

Localisation : Collection 1er étage

Miklos Csörgo # méthode asymptotique # probabilités # statistiques # processus planaire # loi des grands nombres # série temporelle # processus stochastique

60-02 ; 62-02 ; 60F05 ; 60F15 ; 60F17 ; 60G15 ; 60G17 ; 60G50 ; 60G55 ; 60J55 ; 60J65 ; 60K37 ; 62G30 ; 62M10

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- 360 p.
ISBN 978-2-7261-1261-8

Discrete mathematics & theoretical computer science

Localisation : Colloque 1er étage (PARI)

combinatoire # marche aléatoire

05-06 ; 60G50

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- vi; 162 p.
ISBN 978-0-8218-4649-0

Contemporary mathematics , 0485

Localisation : Collection 1er étage

théorie ergodique

28D05 ; 34C28 ; 37A05 ; 37A20 ; 37A45 ; 42A16 ; 47A35 ; 60F15 ; 60G50 ; 62J05 ; 37-06 ; 37Axx ; 00B25

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- 520 p.
ISBN 978-285629-224-2

Astérisque , 0307

Localisation : Périodique 1er étage

variété kählérienne # métrique extrémale # stabilité # dynamique holomorphe # mesure d'équilibre # ensemble exceptionnel# entropie # automorphe # formule de trace # endoscopie # lemme fondamental # G-torseurs # formes quadratiques # cycles algébriques # motifs # variété de Shimira # variété modulaire # sous-variété # correspondance de Hecke # variété hyperkählerienne # cône ample # cône neuf # cône pseudo-effectif # classes grandes # cône de Kähler # courant # métrique singulière # décomposition de Zariski # volume d'un fobré en droites # variété uniréglée # courbe mobile # progression arithmétique # nombres premiers # conjecture de Mumford # espace de module des courbes # groupe modulaire de Teichmuller # théorie de Morse # stratifactionn # catégorie dérivée # catégorie triangulée # variété de Calabi-Yau # flop # variété de dimension 3 # conjecture de Poincaré # flot de Ricci # verre de spin # modèle de Sherrington-Kirkpatrick # énergie libre # brisure de la symétrie des répliques # algèbres simples centrales # indice # exposant # corps de fonctions de deux variables # surfaces complexes # groupe de Brauer # algèbres d'Azumaya # fibrés vectoriels # transformation élémentaire # déformation # géométrie conforme # dimension 4 # théorème de pincement # théorème de la sphère # paires conformes # opérateur de Paneitz # Q-courbure # problèmes de recouvrement # point favori # point épais # point fin # point tardif # analyse multi-fractale # mesure d'occupation # arbre # marche aléatoire # mouvement brownien variété kählérienne # métrique extrémale # stabilité # dynamique holomorphe # mesure d'équilibre # ensemble exceptionnel# entropie # automorphe # formule de trace # endoscopie # lemme fondamental # G-torseurs # formes quadratiques # cycles algébriques # motifs # variété de Shimira # variété modulaire # sous-variété # correspondance de Hecke # variété hyperkählerienne # cône ample # cône neuf # cône pseudo-effectif # classes grandes # cône de ...

35Q15 ; 53D20 ; 53C55 ; 32H50 ; 14D20 ; 20G30 ; 11E04 ; 14C25 ; 11G18 ; 14G35 ; 32J27 ; 14M20 ; 14E30 ; 14C20 ; 14C17 ; 14C30 ; 32C30 ; 11N13 ; 11B25 ; 32G15 ; 57R20 ; 55R40 ; 55R65 ; 55P15 ; 14Exx ; 14Jxx ; 18Exx ; 57N10 ; 53C44 ; 58J35 ; 82B44 ; 60K37 ; 14F22 ; 14F05 ; 14B12 ; 16K50 ; 14G99 ; 53C21 ; 53C20 ; 58J60 ; 58J05 ; 35J60 ; 60G50 ; 60J65 ; 60J55 ; 28A80

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- 145 p.
ISBN 978-0-8218-3869-3

Contemporary mathematics , 0430

Localisation : Collection 1er étage

théorie ergodique # système dynamique @ transformation conservant la mesure # équivalence d'orbite # transformation de Hilbert

28D05 ; 37A05 ; 37A20 ; 37A50 ; 47A35 ; 47A16 ; 60F15 ; 82C20 ; 60G50

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- 272 p.
ISBN 978-0-8218-3466-4

Contemporary mathematics , 0336

Localisation : Collection 1er étage

analyse stochastique # modèle stochastique # processus stochastique # mathématique financière # intégrale stochastique # équilibre # entropie # risque de crédit # EDP stochastique

60-06 ; 91-06 ; 60E15 ; 60F10 ; 60G15 ; 60G50 ; 60H05 ; 60H10 ; 60H15 ; 60J60 ; 91B26 ; 91B30

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- 466 p.
ISBN 978-3-540-43736-9

Lecture notes in mathematics , 1781

Localisation : Collection 1er étage

probabilité # statistique # grande déviation # marche aléatoire # superprocessus de Dawson-Watanabe # statistique semi-paramétrique # estimation # propriété asymptotique

60-06 ; 62-06 ; 60F10 ; 60G57 ; 60G50 ; 60K35 ; 62G05

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ISBN 978-3-540-08061-9

Lecture notes in mathematics , 0566

Localisation : Collection 1er étage

accroissement de Erdös-Renyi # alternative contigue de processus empirique # approche D indice k # approximation faible # boule en espace de Banach # convergence d'écart isotrope # convergence faible # convergence faible vers loi stable # convergence uniforme de mesure # distribution empirique # estimation de paramètre # fonction de distribution empirique # module de continuité de Lévy # principe d'invariance faible # principe d'invariance presque sûre # processus de Kiefer # processus empirique multivarié # test de Kolmogorov-Smirnov # théorème de Glivenko-Cantelli # variable aléatoire faiblement dépendante accroissement de Erdös-Renyi # alternative contigue de processus empirique # approche D indice k # approximation faible # boule en espace de Banach # convergence d'écart isotrope # convergence faible # convergence faible vers loi stable # convergence uniforme de mesure # distribution empirique # estimation de paramètre # fonction de distribution empirique # module de continuité de Lévy # principe d'invariance faible # principe d'invariance ...

60B10 ; 60Fxx ; 60G50 ; 62D05 ; 62E20

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ISBN 978-0-8176-3887-0

Progress in probability , 0038

Localisation : Colloque 1er étage (SILI)

analyse stochastique # espace de Poisson # espace de mesure # espace topologique linéaire # formule de Feymann-Kac # stochastique elliptique # variable aléatoire # équation différentielle # équation différentielle stochastique # équation stochastique de la chaleur

58G32 ; 60B11 ; 60G50 ; 60H10 ; 60Hxx

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- 222 p.
ISBN 978-0-8218-3118-2

Proceedings of the Steklov institute of mathematics , 0174

Localisation : Collection 1er étage

probabilité # théorème limite # variable aléatoire

60-02 ; 60E07 ; 60E15 ; 60F05 ; 60G50

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- 409 p.
ISBN 978-0-387-15203-5

Lecture notes in mathematics , 1117

Localisation : Collection 1er étage

60-02 ; 60F05 ; 60G46 ; 60G50 ; 62A05

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Research talks

20B30 ; 60G50 ; 05C81

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Research schools

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé-Galton-Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé-Galton-Watson trees.
Section 2 defines Bienaymé-Galton-Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.
We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms ...

60J80 ; 60G50 ; 05C05

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Research schools

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé-Galton-Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé-Galton-Watson trees.
Section 2 defines Bienaymé-Galton-Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.
We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms ...

60J80 ; 60G50 ; 05C05

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research schools

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé-Galton-Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé-Galton-Watson trees.
Section 2 defines Bienaymé-Galton-Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.
We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms ...

60J80 ; 60G50 ; 05C05

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Research schools;Probability and Statistics

We first introduce the Metropolis-Hastings algorithm. We then consider the Random Walk Metropolis algorithm on $R^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one dimensional law. It is well-known that, in the limit $n$ tends to infinity, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained for each component of the Markov chain. We generalize this result when the initial distribution is not the target probability measure. The obtained diffusive limit is the solution to a stochastic differential equation nonlinear in the sense of McKean. We prove convergence to equilibrium for this equation. We discuss practical counterparts in order to optimize the variance of the proposal distribution to accelerate convergence to equilibrium. Our analysis confirms the interest of the constant acceptance rate strategy (with acceptance rate between 1/4 and 1/3). We first introduce the Metropolis-Hastings algorithm. We then consider the Random Walk Metropolis algorithm on $R^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one dimensional law. It is well-known that, in the limit $n$ tends to infinity, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained ...

60J22 ; 60J10 ; 60G50 ; 60F17 ; 60J60 ; 60G09 ; 65C40 ; 65C05

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