Programme de la journée 2017

8h30 -- 9h00 Café d'accueil.
9h00 -- 9h20 Présentation de la journée par les organisateurs.
9h20 -- 10h20 Dmitry Chelkak 2D Ising model: combinatorics, CFT/CLE description at criticality and beyond
10h20 -- 10h40 Pause café.
10h40 -- 11h00 Paul Melotti Récurrence spatiales, modèles associés et leurs formes limites Slides
11h00 -- 11h20 Thomas Budzinski Flips sur les triangulations de la sphere : une borne inférieure pour le temps de mélange Slides
11h20 -- 11h40 Gabriela Ciolek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains Slides
11h40 -- 12h00 Simon Coste Trou spectral de matrices de Markov sur des graphes Slides
12h00 -- 12h20 Alkéos Michaïl Perturbations of a large matrix by random matrices Slides
12h40 -- 13h40 Pause déjeuner
13h40 -- 14h00 Léo Miolane Limites fondamentales pour l'estimation de matrices de petit rang Slides
14h00 -- 14h20 Perla El Kettani A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion Slides
14h20 -- 14h40 Julie Fournier Identification and isotropy characterization of deformed random fields through excursion sets Slides
14h40 -- 15h00 Henri Elad Altman Formules de Bismut Elworthy Li pour les processus de Bessel Slides
15h00 -- 15h20 Mohamed Ndaoud Constructing the fractional Brownian motion Slides
15h20 -- 15h40 Marion Sciauveau Cost functionals for large random trees Slides
15h40 -- 16h00 Pause café.
16h00 -- 16h20 Raphael Forien Flux de genes a travers une barriere geographique Slides
16h20 -- 16h40 Veronica Miro Pina Chromosome painting Slides
16h40 -- 17h40 Remco van der Hofstad Hypercube percolation

Les résumés des exposés de la journée

9h20--10h20 Dmitry Chelkak (Russian Academy of Science and ENS)

2D 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.

10h40--11h00 Paul Melotti (UPMC)

Récurrence 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.

11h20--11h40 Thomas Budzinski (Universite Paris-Saclay et ENS)

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

11h20--11h40 Gabriela Ciolek (Telecom ParisTech)

Sharp 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.

11h40--12h00 Simon Coste (Universite Paris-Diderot et Universite Paul Sabatier)

Trou 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.

12h00--12h20 Alkéos Michaïl (Universite Paris Descartes)

Perturbations 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.

13h40--14h00 Léo Miolane (INRIA et ENS)

Phase transitions in low-rank matrix estimation. (joint work with Marc Lelarge)

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.

14h00--14h20 Perla El Kettani (Universite Paris-Sud)

A 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.

14h20--14h40 Julie Fournier (Universite Paris-Descartes & UPMC)

Identification 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.

14h40 -- 15h00 Henri Elad Altman (UPMC)

Formules 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.

15h00--15h20 Mohamed Ndaoud (X-CREST)

Constructing 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.

15h20--15h40 Marion Sciauveau (Ecole des Ponts)

Cost 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.

16h00--16h20 Raphael Forien (Ecole Polytechnique)

Gene 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.

16h20 -- 16h40 Veronica Miro Pina (UPMC)

Chromosome 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.

16h40--17h40 Remco van der Hofstad (Technische Universiteit Eindhoven)

Hypercube 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.