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Documents  Miermont, Grégory | enregistrements trouvés : 6

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Martingales in self-similar growth-fragmentations and their applications Bertoin, Jean | CIRM H

Post-edited

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 ...

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Vertex degrees in planar maps Drmota, Michael | CIRM H

Multi angle

Combinatorics;Probability and Statistics

We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner
We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even inf...

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Nesting statistics in the $O(n)$ loop model on random planar maps Bouttier, Jérémie | CIRM H

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Combinatorics;Mathematical Physics;Probability and Statistics

The $O(n)$ model can be formulated in terms of loops living on the lattice, with n the fugacity per loop. In two dimensions, it is known to possess a rich critical behavior, involving critical exponents varying continuously with n. In this talk, we will consider the case where the model is ”coupled to 2D quantum gravity”, namely it is defined on a random map.
It has been known since the 90’s that the partition function of the model can be expressed as a matrix integral, which can be evaluated exactly in the planar limit. A few years ago, together with G. Borot and E. Guitter, we revisited the problem by a combinatorial approach, which allows to relate it to the so-called Boltzmann random maps, which have no loops but faces of arbitrary (and controlled) face degrees. In particular we established that the critical points of the $O(n)$ model are closely related to the ”stable maps” introduced by Le Gall and Miermont.
After reviewing these results, I will move on to a more recent work done in collaboration with G. Borot and B. Duplantier, where we study the nesting statistics of loops. More precisely we consider loop configurations with two marked points and study the distribution of the number of loops separating them. The associated generating function can be computed exactly and, by taking asymptotics, we show that the number of separating loops grows logarithmically with the size of the maps at a (non generic) critical point, with an explicit large deviation function. Using a continuous generalization of the KPZ relation, our results are in full agreement with those of Miller, Watson and Wilson concerning nestings in Conformal Loop Ensembles.
The $O(n)$ model can be formulated in terms of loops living on the lattice, with n the fugacity per loop. In two dimensions, it is known to possess a rich critical behavior, involving critical exponents varying continuously with n. In this talk, we will consider the case where the model is ”coupled to 2D quantum gravity”, namely it is defined on a random map.
It has been known since the 90’s that the partition function of the model can be ...

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Vanishing corrections for the position of an FKPP front Berestycki, Julien | CIRM H

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Partial Differential Equations;Probability and Statistics

The celebrated Fisher-Kolmogorov-Petrovsky-Piscounof equation (FKPP) in one dimension for
$h:\mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}$ is:

$\partial_th = \partial{_x^2}h + h - h^2, h(x, 0) = h_0(x)$.

This equation is a natural description of a reaction-diffusion model (Fisher 1937, Kolmogorov et al. 1937, Aronson 1978). It is also related to branching Brownian motion: for the Heaviside initial condition $h_0 (x) = 1{_x<0}$ , $h(x, t)$ is the probability that the rightmost particle at time t in a branching Brownian motion (BBM) is to the right of $x$.
One of the beauty of this equation is that for initial conditions that decrease sufficiently fast, a front develops, i.e. there exists a centring term $m(t)$ and an asymptotic shape $\omega(x)$ such that

$\lim_{t \to \infty} h(m(t) + x,t) = \omega(x) \in (0, 1).$

Since the original paper of Kolmogorov et al., the position of the front $m(t)$ has been studied intensely, in particular by Bramson. In this talk, I will present some recent results concerning a prediction of Ebert and van Saarloos about the vanishing corrections of this position.
Based on a joint work with E. Brunet.
The celebrated Fisher-Kolmogorov-Petrovsky-Piscounof equation (FKPP) in one dimension for
$h:\mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}$ is:

$\partial_th = \partial{_x^2}h + h - h^2, h(x, 0) = h_0(x)$.

This equation is a natural description of a reaction-diffusion model (Fisher 1937, Kolmogorov et al. 1937, Aronson 1978). It is also related to branching Brownian motion: for the Heaviside initial condition $h_0 (x) = 1{_x<0}$ , $h(x, t)$ is ...

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Recurrence of half plane maps Angel, Omer | CIRM H

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Combinatorics;Probability and Statistics

On a graph $G$, we consider the bootstrap model: some vertices are infected and any vertex with 2 infected vertices becomes infected. We identify the location of the threshold for the event that the Erdos-Renyi graph $G(n, p)$ can be fully infected by a seed of only two infected vertices. Joint work with Brett Kolesnik.

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Quelques interactions entre analyse, probabilités et fractals Barral, J. ; Berestycki, Julien ; Bertoin, J. ; Fan, A. H. ; Haas, B. ; Jaffard, S. ; Miermont, Grégory ; Peyrière, J. | Société Mathématique de France 2010

Ouvrage

- x; 243 p.
ISBN 978-2-85629-313-3

Panoramas et synthèses , 0032

Localisation : Collection 1er étage

Approximation diophantienne # arbres aléatoires # cascade multiplicative # chaos multiplicatif # chaîne de Markov # dimension de boîte # dimension de Hausdorff # dimension de packing # fonction multifractale # formalisme multifractal # fractals # fragmentation aléatoire # martingales # mesure multifractale # mesures # processus de branchement # produits de Riesz # recouvrements # régularité ponctuelle # spectre multifractal # systèmes dynamiques # ubiquité Approximation diophantienne # arbres aléatoires # cascade multiplicative # chaos multiplicatif # chaîne de Markov # dimension de boîte # dimension de Hausdorff # dimension de packing # fonction multifractale # formalisme multifractal # fractals # fragmentation aléatoire # martingales # mesure multifractale # mesures # processus de branchement # produits de Riesz # recouvrements # régularité ponctuelle # spectre multifractal # systèmes dynamiques ...

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