Les exposés de la deuxième journée

La deuxième journée des probabilités de demain s'est déroulée à l'IHÉS le 11 mai 2017. Les vidéos sont disponibles à cette adresse, ou ci-dessous.

Présentation de la journée par les organisateurs. Vidéo
Dmitry Chelkak
2D Ising model: combinatorics, CFT/CLE description at criticality and beyond2D Ising model: combinatorics, CFT/CLE description at criticality and beyond

We begin this expository talk with a discussion of the combinatorics of the nearest-neighbor Ising model in 2D - an archetypical example of a statistical physics system that admits an order-disorder phase transition - and the underlying fermionic structure, which makes it accessible for the rigorous mathematical analysis. We then survey recent results on convergence of correlation functions at the critical temperature to conformally covariant scaling limits given by Conformal Field Theory, as well as the convergence of interfaces (domain walls) to the relevant Conformal Loop Ensemble. Is the case closed? Not at all: there are still many things to understand and to prove, especially for the non-critical and/or non-homogeneous model.

Vidéo
Paul Melotti
Récurrence spatiales, modèles associés et leurs formes limitesRécurrence spatiales, modèles associés et leurs formes limites

Certaines relations polynomiales, telles que les relations vérifiées par les mineurs d'une matrice, peuvent être interprétées comme des relations de récurrence sur Z^3. Dans certains cas, les solutions de ces récurrences présentent une propriété inattendue : ce sont des polynômes de Laurent en les conditions initiales. Peut-on donner une interprétation combinatoire de ce fait ? On verra que lorsqu'un objet combinatoire caché derrière ces relations est identifié, il présente des phénomènes de formes limites qui peuvent être calculées explicitement, le plus connu étant le "cercle arctique" des pavages du diamant aztèque. On parlera des récurrences dites de l'octaèdre, du cube, et d'une récurrence due à Kashaev.

Slides Vidéo
Thomas Budzinski
Flips sur les triangulations de la sphere : une borne inférieure pour le temps de mélangeFlips sur les triangulations de la sphere : une borne inférieure pour le temps de mélange

One of the simplest ways to sample a uniform triangulation of the sphere with a fixed number n of faces is a Monte-Carlo method: we start from an arbitrary triangulation and flip repeatedly a uniformly chosen edge, i.e. we delete it and replace it with the other diagonal of the quadrilateral that appears. We will prove a lower bound of order n^{5/4} on the mixing time of this Markov chain.

Slides Vidéo
Gabriela Ciolek
Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chainsSharp Bernstein and Hoeffding type inequalities for regenerative Markov chains

The purpose of this talk is to present Bernstein and Hoeffding type functional inequalities for regenerative Markov chains. Furthermore, we generalize these results and show exponential bounds for suprema of empirical processes over a class of functions F which size is controlled by its uniform entropy number. All constants involved in the bounds of the considered inequalities are given in an explicit form which can be advantageous in practical considerations. We present the theory for regenerative Markov chains, however the inequalities are also valid in the Harris recurrent case.

Slides Vidéo
Simon Coste
Trou spectral de matrices de Markov sur des graphesTrou spectral de matrices de Markov sur des graphes

La théorème d’Alon-Friedman dit que la deuxième valeur propre d’un graphe aléatoire d-régulier converge vers 2sqrt(d-1) lorsque la taille du graphe tend vers l’infini. Ce théorème difficile est relié à des propriétés essentielles du graphe G, comme par exemple sa constante d’expansion ou encore la vitesse de convergence de la marche aléatoire simple sur G. Dans cet exposé, nous présenterons ces liens entre la deuxième valeur propre et les propriétés des graphes réguliers puis nous généraliserons ces résultats à des modèles de graphes plus généraux, en particulier des graphes orientés.

Slides Vidéo
Alkéos Michaïl
Perturbations of a large matrix by random matricesPerturbations of a large matrix by random matrices

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear (called perturbative and semi-perturbative regimes): the leading terms of the expansion are either related to free probability theory or to the one-dimensional Gaussian free field.

Slides Vidéo
Léo Miolane
Limites fondamentales pour l'estimation de matrices de petit rangLimites fondamentales pour l'estimation de matrices de petit rang

We consider the estimation of noisy low-rank matrices. Our goal is to compute the minimal mean square error (MMSE) for this statistical problem. We will observe a phase transition: there exists a critical value of the signal-to-noise ratio above which it is possible to make a non-trivial guess about the signal, whereas this is impossible below this critical value.

Slides
Perla El Kettani
A stochastic mass conserved reaction-diffusion equation with nonlinear diffusionA stochastic mass conserved reaction-diffusion equation with nonlinear diffusion

In this talk, we study a stochastic mass conserved reaction-diffusion equation with a linear or nonlinear diffusion term and an additive noise corresponding to a Q-Brownian motion. We prove the existence and the uniqueness of the weak solution. The proof is based upon the monotonicity method. This is joint work with D.Hilhorst and K.Lee.

Slides Vidéo
Julie Fournier
Identification and isotropy characterization of deformed random fields through excursion setsIdentification and isotropy characterization of deformed random fields through excursion sets

A deterministic application θ : R² → R² deforms bijectively and regularly the plane and allows to build a deformed random field X ◦ θ : R² → R from a regular, stationary and isotropic random field X : R² → R. The deformed field X ◦ θ is in general not isotropic, however we give an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X ◦ θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. Deformed fields satisfying this property are proved to be strictly isotropic. Besides, assuming that the mean Euler characteristic of excursions sets of X ◦ θ over some basic domains is known, we are able to identify θ.
Reference: hal-01495157.

Slides Vidéo
Henri Elad Altman
Formules de Bismut Elworthy Li pour les processus de BesselFormules de Bismut Elworthy Li pour les processus de Bessel

Bessel processes are a one-parameter family of nonnegative diffusion processes with a singular drift. When the parameter (called dimension) is smaller than one, the drift is non-dissipative, and deriving regularity properties for the transition semigroup in such a regime is a very difficult problem in general.
In my talk I will show that, nevertheless, the transition semigroups of Bessel processes of dimension between 0 and 1 satisfy a Bismut-Elworthy-Li formula, with the particularity that the martingale term is only in L^{p} for some p > 1, rather than L^{2} as in the dissipative case. As a consequence some interesting strong Feller bounds can be obtained.

Slides Vidéo
Mohamed Ndaoud
Constructing the fractional Brownian motionConstructing the fractional Brownian motion

In this talk, we give a new series expansion to simulate B a fractional Brownian motion based on harmonic analysis of the auto-covariance function. The construction proposed here reveals a link between Karhunen-Loève theorem and harmonic analysis for Gaussian processes with stationarity conditions. We also show some results on the convergence. In our case, the convergence holds in L2 and uniformly, with a rate-optimal decay of the norm of the rest of the series in both senses.

Slides Vidéo
Marion Sciauveau
Cost functionals for large random treesCost functionals for large random trees

Les arbres apparaissent naturellement dans de nombreux domaines tels que l'informatique pour le stockage de données ou encore la biologie pour classer des espèces dans des arbres phylogénétiques. Dans cet exposé, nous nous intéresserons aux limites de fonctionnelles additives de grands arbres aléatoires. Nous étudierons le cas des arbres binaires sous le modèle de Catalan (arbres aléatoires choisis uniformément parmi les arbres binaires enracinés complets ordonnés avec un nombre de nœud donné). On obtiendra un principe d'invariance pour ces fonctionnelles ainsi que les fluctuations associées.
La preuve repose sur le lien entre les arbres binaires et l'excursion brownienne normalisée.

Slides Vidéo
Raphael Forien
Gene Flow across a geographical barrierGene Flow across a geographical barrier

Consider a species scattered along a linear habitat. Physical obstacles can locally reduce migration and genetic exchanges between different parts of space. Tracing the position of an individual's ancestor(s) back in time allows to compute the expected genetic composition of such a population. These ancestral lineages behave as simple random walks on the integers outside of a bounded set around the origin. We present a continuous real-valued process which is obtained as a scaling limit of these random walks, and we give several other constructions of this process.

Slides Vidéo
Veronica Miro Pina
Chromosome paintingChromosome painting

We consider a simple population genetics model with recombination. We assume that at time 0, all individuals of a haploid population have their unique chromosome painted in a distinct color. At rare birth events, due to recombination (modeled as a single crossing-over), the chromosome of the newborn is a mosaic of its two parental chromosomes. The partitioning process is then defined as the color partition of a sampled chromosome at time t. When t is large, all individuals end up having the same chromosome.
I will discuss some results on the partitioning process at stationarity, concerning the number of colours and the description of a typical color cluster.

Slides Vidéo
Remco van der Hofstad
Hypercube percolationHypercube percolation

Consider bond percolation on the hypercube {0,1}^n at the critical probability p_c defined such that the expected cluster size equals 2^{n/3}, where 2^{n/3} acts as the cube root of the number of vertices of the n-cube. Percolation on the Hamming cube was proposed by Erdös and Spencer (1979), and has proved to be substantially harder than percolation on the complete graph. In this talk, I will describe the percolation phase transition on the hypercube, and show that it shares many features with that on the complete graph.
In previous work with Borgs, Chayes, Slade and Spencer, and with Heydenreich, we have identified the subcritical and critical regimes of percolation on the hypercube. In particular, we know that for p=p_c(1+O(2^{-n/3})), the largest connected component has size roughly 2^{2n/3} and that this quantity is non-concentrated. In work with Asaf Nachmias, we identify the supercritical behavior of percolation on the hypercube, by showing that, for any sequence \epsilon_n tending to zero, but \epsilon_n being much larger than 2^{-n/3}, percolation at p_c(1+\epsilon_n) has, with high probability, a unique giant component of size (2+o(1))\epsilon_n 2^n. This also confirms that the validity of the proposed critical value. Finally, we `unlace' the proof by identifying the scaling of component sizes in the supercritical and critical regimes without relying on the percolation lace expansion. The lace expansion is a beautiful technique that is the major technical tool for high-dimensional percolation, but that is also quite involved and can have a disheartening effect on some

Vidéo

Les vidéos de la journée

Dmitry Chelkak (Russian Academy of Science and ENS)

2D Ising model: combinatorics, CFT/CLE description at criticality and beyond

Paul Melotti (UPMC)

Récurrence spatiales, modèles associés et leurs formes limites

Thomas Budzinski (Universite Paris-Saclay et ENS)

Flip sur les triangulations de la sphere : une borne inférieure pour les temps de mélange

Gabriela Ciolek (Telecom ParisTech)

Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains

Simon Coste (Universite Paris-Diderot et Universite Paul Sabatier)

Trou spectral de matrices de Markov sur des graphes

Alkéos Michaïl (Universite Paris Descartes)

Perturbations of a large matrix by random matrices

Perla El Kettani (Universite Paris-Sud)

A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion

Julie Fournier (Universite Paris-Descartes & UPMC)

Identification and isotropy characterization of deformed random fields through excursion sets

Henri Elad Altman (UPMC)

Formules de Bismut Elworthy Li pour les processus de Bessel

Mohamed Ndaoud (X-CREST)

Constructing the fractional Brownian motion

Marion Sciauveau (Ecole des Ponts)

Cost functionals for large random trees

Raphael Forien (Ecole Polytechnique)

Gene Flow across a geographical barrier

Veronica Miro Pina (UPMC)

Chromosome painting

Remco van der Hofstad (Technische Universiteit Eindhoven)

Hypercube percolation