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We estimate the lowest eigenvalue in the gap of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schrödinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. Most of our result are established in the Birman-Schwinger reformulation of the problem.

Chapters 2-12 are based on published or submitted works - see the corresponding links in the metadata. Some of them are joint work with Laure Marêché, Fabio Martinelli, Tamás Mezei, Robert Morris, Réka Szabó and Cristina Toninelli; We study two tightly related classes of statistical mechanics models on the two-dimensional square lattice—kinetically constrained models and bootstrap percolation. The former arose as models of the dynamics of supercooled liquids close to the glass transition, while the latter are used to model a number of settings including magnets and social phenomena. We consider both kinetically constrained models and bootstrap percolation from a rigorous probabilistic perspective. We are interested in their behaviour as their parameter approaches its (possibly degenerate) critical value. More specifically, we investigate the rate of divergence of certain characteristic time scales, such as the infection time of a fixed site and the relaxation time.Among the highlights of the thesis is determining the universality classes of kinetically constrained models together with their characteristic equilibrium time scales at low temperature. That is, we establish a partition of all possible models into groups with similar behaviour and provide a recipe for determining the behaviour from the definition of the model. Contributions are made to the full spectrum of universality classes of kinetically constrained models, but in some cases also to the simpler and better understood bootstrap percolation. In addition to universal results, we provide sharp asymptotics in both bootstrap percolation and kinetically constrained models for the most classical, 2-neighbour, model, as well as advances on the 1-neighbour kinetically constrained model known as the Fredrickson–Andersen 1-spin facilitated model.The thesis consists of three main parts based on techniques from different domains. The first one relates to dynamics of interacting particle systems. The second one relies on combinatorial arguments. The third and final part takes a percolation viewpoint.; On étudie deux classes de modèles étroitement liées de physique statistique sur le réseau carré bidimensionnel – les modèles cinétiquement contraints et la percolation bootstrap. Les premiers sont apparus pour modéliser la dynamique des liquides surfondus près de leur transition vitreuse, tandis que la percolation bootstrap modélise de nombreux cadres tels que certains aimants ou encore des phénomènes sociaux. Nous considérons les modèles cinétiquement contraints et la percolation bootstrap d’un point de vue rigoureux probabiliste. On s’intéresse à leur comportement lorsque leur paramètre tend vers sa valeur critique (possiblement dégénérée). Plus concrètement, nous étudions le taux de divergence de certains temps caractéristiques tels que le temps d’infection d’un site fixé et le temps de relaxation.Parmi les résultats les plus conséquents de la thèse est la détermination des classes d’universalité de modèles cinétiquement contraints ainsi que leurs échelles de temps caractéristiques à l’équilibre en basse température. C’est-à-dire, on établit une partition de tous les modèles possibles en groupes à comportement similaire et fournit une recette pour déterminer ce comportement à partir de la définition du modèle. Des contributions sont apportées à tout le spectre de classes d’universalité de modèles cinétiquement contraints, mais dans certains cas aussi à la percolation bootstrap plus simple et mieux comprise. En supplément des résultats universels, nous donnons des asymptotiques exactes à la fois en percolation bootstrap et en modèle cinétiquement contraint pour le modèle le plus classique à deux voisins. De plus, nous marquons des progrès sur le modèle cinétiquement contraint à un voisin appelé modèle de Fredrickson–Andersen 1-spin facilité.La thèse est constituée de trois parties principales, basées sur des techniques provenant de domaines différents. La première relève de la dynamique de systèmes de particules en interaction. La deuxième emploie des arguments de combinatoire. La troisième et dernière partie prend un point de vue de percolation.

Ce document contient hal-02887010, v1: Stability in Gagliardo-Nirenberg inequalities and hal-02887013, v1: Stability in Gagliardo-Nirenberg inequalities. Supplementary material, avec plusieurs additions dont: Chapitre 1 (variational methods) et Chapitre 6 (Sobolev's inequality).; International audience; The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods.

Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of this work is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we are particularly interested in is that of having either rapidly decaying or long-range correlations. In a nutshell, our main result is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. We have two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap. In all these statements, our aim is to get a quantitative estimate of the considered quantity (generic correlation length or spectral gap), which has the best possible dependency on the physical and bond dimensions of the random MPS or PEPS., 53 pages, 7 figures. v2: a bug in Section 4 has been fixed and a few changes have been made to the presentation. v3: published version

International audience; Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropyentropy production inequalities, with applications to stability in Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

Several techniques of generating random quantum channels, which act on the set of $d$-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show under which conditions they become mathematically equivalent, and lead to the uniform, Lebesgue measure on the convex set of quantum operations. We compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity and the $2$-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed and their spectral properties are studied using the Bloch representation of a classical probability vector., 29 pages, 7 figures

International audience; This paper considers the penalized least squares estimators with convex penalties or regularisation norms. We provide sparsity oracles inequalities for the prediction error for a general convex penalty and for the particular cases of Lasso and Group Lasso estimators in a regression setting. The main contributions are that our oracle inequalities are established for the more general case where the observations noise is issued from probability measures that satisfy a weak spectral gap (or Poincaré) inequality instead of gaussian distributions, and five easier to verify bounds on compatibility. We Illustrate our results on a heavy tailed example and a sub gaussian one; we especially give the explicit bounds of the oracle inequalities for these two special examples.

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods., Comment: This manuscript merges 2007.03674: Stability in Gagliardo-Nirenberg inequalities and 2007.03419: Stability in Gagliardo-Nirenberg inequalities. Supplementary material with several additions including: Chapter 1 (variational methods) and Chapter 6 (Sobolev's inequality)

Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb{Z}^d$ with continuous-time constrained Glauber dynamics. They are a natural non-monotone stochastic version of the family of cellular automata with random initial state known as $\mathcal{U}$-bootstrap percolation. KCM have an interest in their own right, owing to their use for modelling the liquid-glass transition in condensed matter physics. In two dimensions there are three classes of models with qualitatively different scaling of the infection time of the origin as the density of infected sites vanishes. Here we study in full generality the class termed `critical'. Together with the companion paper by Martinelli and two of the authors we establish the universality classes of critical KCM and determine within each class the critical exponent of the infection time as well as of the spectral gap. In this work we prove that for critical models with an infinite number of stable directions this exponent is twice the one of their bootstrap percolation counterpart. This is due to the occurrence of `energy barriers', which determine the dominant behaviour for these KCM but which do not matter for the monotone bootstrap dynamics. Our result confirms the conjecture of Martinelli, Morris and the last author, who proved a matching upper bound., 33 pages, 6 figures, expanded section 5.1, minor improvements of the presentation

International audience; By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor $2$. The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.

The purpose of this paper is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg inequalities. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. In both cases, the tail behaviour plays a key role. The regularity properties of the parabolic flow allow us to connect an improved entropy-entropy production inequality during the initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymp-totic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms the distance to the manifold of optimal functions.

We call pattern any non-constant stable solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [19] and Matano [49] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the magnitude of the nonlinearity, and the stability of patterns. We propose several applications, for example, we prove that (under a geometric assumption) there exist no patterns if the domain is shrunk or if the nonlinearity has a small magnitude. We also refine the result of Casten Holland and Matano and show that it is robust under smooth perturbations of the domain and the nonlinearity. In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincar{\'e} inequality), stating that the L2-norm of the gradient of a solution is controlled by the L2-norm of f(u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates. Our approach relies on the introduction of what we call the Robin-curvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions. Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [53] and to extend some results on De Giorgi's conjecture to a larger class of domains.

International audience; Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the V-geometrical ergodicity of a general Markov kernel P , which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of P by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in V-geometrical ergodicity.

The purpose of this Ph.D. thesis is to study precise large and moderate deviation asymptotics for products of independent and identically distributed random matrices. In the first part, we establish Bahadur-Rao type and Petrov type exact asymptotics of large deviation probabilities for the norm cocycle «dollard»\log|G_nx|»dollard», where «dollard»G_n = g_n\ldots g_1»dollard» is the product of independent and identically distributed random «dollard»d\times d»dollard» matrices «dollard»g_i»dollard», «dollard»x»dollard» is a unit vector in «dollard»\mathbb R^d»dollard». The second part is devoted to establishing Bahadur-Rao type and Petrov type large deviations for the «dollard»(i,j)»dollard»-th entries «dollard»G_n^{i,j}»dollard» of «dollard»G_n»dollard». In particular, our result improves significantly the large deviation bounds established recently. In the third part, we investigate the Berry-Esseen bound and Cramér type moderate deviation expansion for the norm cocycle of products of random matrices. These results are proved by elaborating a new approach based on a smoothing inequality in the complex plane and on the saddle point method. The fourth part is devoted to studying Berry-Esseen bounds and Cramér type moderate deviation expansions for the operator norm «dollard»\|G_n\|»dollard», the entries «dollard»G_n^{i,j}»dollard» and the spectral radius «dollard»\rho(G_n)»dollard», for positive matrices. In the fifth part, we study the Berry-Esseen type bounds and moderate deviation principles for the operator norm «dollard»\|G_n\|»dollard» and the spectral radius «dollard»\rho(G_n)»dollard», for invertible matrices. We also prove the moderate deviation expansions in the normal range «dollard»[0, o(n^{1/6})]»dollard». The sixth part is devoted to the Cramér type moderate deviation expansion for the entries «dollard»G_n^{i,j}»dollard» of products of invertible matrices., L'objet de cette thèse est d'étudier les asymptotiques précises de grandes déviations et de déviations modérées pour les produits de matrices aléatoires indépendantes et identiquement distribuées. Dans la première partie, nous établissons des asymptotiques exactes de types Bahadur-Rao et Petrov pour les probabilités de grandes déviations pour le cocycle de la norme, «dollard»\log|G_nx|»dollard», où «dollard»G_n=g_n\ldots g_1»dollard» est le produit des matrices aléatoires «dollard»g_i»dollard», de type «dollard»d \times d»dollard», indépendantes et identiquement distribuées, «dollard»x»dollard» est un vecteur unitaire de «dollard»\mathbb R^d»dollard». La deuxième partie est consacrée à l'établissement des résultats de grandes déviations de types Bahadur- Rao et Petrov pour les entrées «dollard»(i,j)»dollard»-ème «dollard»G_n^{i,j}»dollard» de «dollard»G_n»dollard». En particulier, notre résultat améliore de manière significative les bornes de grandes déviations établies récemment dans la littérature. Dans la troisième partie, nous obtenons la borne de Berry-Esseen et le développement asymptotique de déviations modérées de type Cramér pour le cocycle de la norme des produits de matrices aléatoires. Ces résultats sont prouvés en élaborant une nouvelle approche basée sur une inégalité de lissage dans le plan complexe et sur la méthode du point-selle. La quatrième partie est consacrée à l'étude des bornes de type Berry-Esseen et au développement asymptotique de déviations modérées de type Cramér pour la norme d'opérateur «dollard»\|G_n\|»dollard», pour les entrées «dollard»G_n^{i,j}»dollard» et le rayon spectral «dollard»\rho(G_n)»dollard» des produits de matrices aléatoires positives. Dans la cinquième partie, nous étudions les bornes de type Berry-Esseen et les principes de déviations modérées pour la norme d'opérateur «dollard»\|G_n\|»dollard» et le rayon spectral «dollard»\rho(G_n)»dollard», pour les matrices inversibles. Nous prouvons également des développements asymptotiques de déviations modérées dans la zone normale «dollard»[0, o (n^{1/6})]»dollard». La sixième partie est consacrée au développement asymptotique de déviation modérée de type Cramér pour les entrées «dollard»G_n^{i,j} «dollard» des produits de matrices aléatoires inversibles.

We give a reformulation of Salem's conjecture about the absence of Salem numbers near one in terms of a uniform spectral gap for certain arithmetic hyperbolic surfaces., Comment: 13 pages

International audience; We call pattern any non-constant stable solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [19] and Matano [49] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the magnitude of the nonlinearity, and the stability of patterns. We propose several applications, for example, we prove that (under a geometric assumption) there exist no patterns if the domain is shrunk or if the nonlinearity has a small magnitude. We also refine the result of Casten Holland and Matano and show that it is robust under smooth perturbations of the domain and the nonlinearity. In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincaré inequality), stating that the L2-norm of the gradient of a solution is controlled by the L2-norm of f(u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates. Our approach relies on the introduction of what we call the Robin-curvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions. Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [53] and to extend some results on De Giorgi's conjecture to a larger class of domains.

We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hard potentials, or hard spheres collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species operator $\mathbf{L}$ is known to possess an explicit spectral gap $\lambda_{\mathbf{L}}$, in the global equilibrium weighted $L^2$ space. We study a new operator $\mathbf{L^\varepsilon}$ obtained by linearizing the Boltzmann operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order $\varepsilon$, $\varepsilon >0$. This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of $\mathbf{L^\varepsilon}$ in the global equilibrium weighted $L^2$ space. More precisely, we consider the explicit upper bound that has been proved for the entropy production functional associated to $\mathbf{L}$ and we show that the same estimate holds for the entropy production functional associated to $\mathbf{L^\varepsilon}$, up to a correction of order $\varepsilon$., Comment: 22 pages

We present three results of Sebastien Gouezel's: the local limit theorem for random walks on hyperbolic groups, a multiplicative ergodic theorem for non-expansive mappings (joint work with Anders Karlsson), and the description of the essential spectrum of the Laplacian on \begin{document}$ SL(2,{\mathbb R}) $\end{document} orbits in the moduli space (joint work with Artur Avila).

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials ("flatness") implies a Ruelle-Perron-Frobenius Theorem and a decay of the transfer operator of the same speed than entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of H{\"o}lder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situation, including Pomeau-Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results., Comment: v3: The published article (Ergodic Theory Dynam. Systems 40, 2020) contained a significant error in Lemma 2.14, used in the core Theorem 4.1. This is a consolidated version of the article, with the error corrected (and a few other minor points improved along the way). In order to fix the error, the assumption on coupling in 4.1 needs to be slightly modified (see also Definition 2.12) and Lemma 2.14 (now numbered 2.15) has been adjusted. Section 5.1 provides a criterion to ensure this new hypothesis in our cases of interest, so that all other results are unaffected. I apologize to readers of the previous version for this embarrassing mistake, and warmly thank Manuel Stadlbauer for pointing out this error to me

Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the $V-$geometrical ergodicity of a general Markov kernel $P$, which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of $P$ by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in $V-$geometrical ergodicity.

The Laughlin state is an ansatz for the ground state of a system of 2D quantum particles submitted to a strong magnetic field and strong interactions. The two effects conspire to generate strong and very specific correlations between the particles. I present a mathematical approach to the rigidity these correlations display in their response to perturbations. This is an important ingredient in the theory of the fractional quantum Hall effect. The main message is that potentials generated by impurities and residual interactions can be taken into account by generating uncorrelated quasi-holes on top of Laughlin's wave-function. An appendix contains a conjecture (not due to me) that should be regarded as a major open mathematical problem of the field, relating to the spectral gap of a certain zero-range interaction., Expository text based on joint works with Elliott H. Lieb, Alessandro Olgiati, Sylvia Serfaty and Jakob Yngvason

Dealing with one-dimensional diffusion operators, we obtain upper and lower variational formulae on the eigenvalues given by the max-min principle, generalizing the celebrated result of Chen and Wang on the spectral gap. Our inequalities reveal to be sharp at least when the eigenvalues considered belong to the discrete spectrum of the operator, since in this case both lower and upper bounds coincide and involve the associated eigenfunctions. Based on the intertwinings between diffusion operators and some convenient gradients with weights, our approach also allows to estimate the gap between the two first positive eigenvalues when the spectral gap belongs to the discrete spectrum.

International audience We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap.

This thesis is devoted to study the large time asymptotic behaviour and hypocoercivity of evolution PDEs. We prove that for Nernst-Planck equation and in several cases of flocking model, there exist optimal exponential rates of convergence to station- ary solutions for large time, and the rates are determined by spectral gap of the linearized problem around the stable solutions. Moreover, for the flocking model, we prove that there exists a threshold value that drives phase transition and classify all station- ary solutions and the linear stability properties. Then for kinetic Fokker-Planck equation, we prove that for its φ-entropy, the exponential rate of decay is faster than the optimal rate up to a zero-measure set in time t. And for linearized Vlasov-Poisson- Fokker-Planck equation with an external potential of confinement, we study the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling.., Cette thèse est consacrée à l’étude du comportement asymptotique dans le temps et de l’hypocoercivité des EDP d’évolution. Nous montrons que pour l’équation de Nernst-Planck et dans les cas spécial de modèle de flocage, il existe des taux de con- vergence exponentiels optimaux vers des solutions stationnaires pour un temps t long, et que ces taux sont déterminés par le trou spectral du problème linéarisé autour des solutions stables. De plus, pour le modèle de flocage, nous prouvons qu’il existe une valeur seuil qui commande la transition de phase et classe toutes les solutions stationnaires et les propriétés de stabilité linéaire. Ensuite, pour l’équation cinétique de Fokker-Planck, nous prouvons que, pour son entropie, le taux de décroissance exponentiel est plus rapide que le taux optimal jusqu’à une mesure nulle définie dans le temps. Et pour l’équation linéarisée de Vlasov-Poisson-Fokker-Planck avec un potentiel de confinement externe, nous étudions le comportement dans le temps long des solutions en utilisant des méthodes d’hypocoercivité et une notion de produit scalaire adaptée à la présence d’un couplage de Poisson.

We show that the tensor product M ⊗ ‾ N of any two full factors M and N (possibly of type III) is full and we compute Connes' invariant τ ( M ⊗ ‾ N ) in terms of τ ( M ) and τ ( N ) . The key novelty is an enhanced spectral gap property for full factors of type III. Moreover, for full factors of type III with almost periodic states, we prove an optimal spectral gap property. As an application of our main result, we also show that for any full factor M and any non-type I amenable factor P, the tensor product factor M ⊗ ‾ P has a unique McDuff decomposition, up to stable unitary conjugacy.

Nous prouvons l’existence de noyaux de Stein pour les mesures de probabilites sur $\mathbb{R}^{d}$ satisfaisant une inegalite de Poincare, et obtenons des bornes sur la discrepance de Stein de telles mesures. Des applications au theoreme central limite sont donnees, dont une nouvelle borne sur la vitesse de convergence en distance de Kantorovitch–Wasserstein $W_{2}$ avec un taux et une dependance en la dimension optimales. Comme corollaire, nous obtenons une version quantitative d’une borne sur la constante de Poincare de mesures de probabilites satisfaisant une contrainte sur le moment d’ordre 2. Les resultats sont plus generalement valides dans le cadre de mesures verifiant une inegalite de Poincare a poids inversee. La preuve est basee sur des arguments simples d’analyse fonctionnelle. De plus, nous demontrons deux proprietes generales sur la discrepance de Stein, valide des lors qu’un noyau de Stein existe : la discrepance de Stein est strictement decroissante le long du TCL, et elle controle le moment d’ordre 3 d’un vecteur aleatoire.

Applying quantitative perturbation theory for linear operators, we prove nonasymptotic bounds for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions $\mathscr{X}$. The main results are concentration inequalities and Berry–Esseen bounds, obtained assuming neither reversibility nor “warm start” hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform $\mathscr{X}$-ergodicity hypothesis, and when $\mathscr{X}$ consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the flexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.

The subject of this thesis is the introduction, the study and the applications of multidimensional complex cones. First, we study the grassmannian of Banach space. We define a notion of right decomposition for p-dimensional spaces and we prove the equivalence between theHausdorff distance on the grassmannian and the distance given by a norm on the exterior algebra.Then, we define p-dimensional complex cones and a gauge on the subspaces of dimension p of these cones. We show a contraction principle for thisgauge. This allows us to prove, for an operator contracting such a cone, the existence of a spectral gap which isolate the p leading eigenvaluesfrom the rest of the spectrum. We use this theory to prove a theorem of analytic regularity for Lyapunov exponents of a random product ofoperators contracting a cone. We also give a comparison between the Hausdorff distance for vector spaces and our gauge.Finally, we introduce a notion of dual cone for p-dimensional cones. In this setting, we prove that the topological properties of a cone translateinto topological properties for its dual and conversely. We complete the previous regularity theorem by proving the existence and the regularity ofa dominated splitting of the space into a "fast space" and a "slow space".; L'objet de cette thèse est l'introduction, l'étude et l'utilisation des cônes complexes multidimensionnels. Dans un premier temps, nous étudions la grassmannienne des espaces de Banach. Nous définissons une notion de bonne décomposition pour les espaces de dimension p et nous démontronsl'équivalence entre la distance de Hausdorff sur la grassmannienne et la distance fournie par une norme sur l'algèbre extérieure.Dans un deuxième temps, nous définissons les cônes complexes p-dimensionnels ainsi qu'une jauge sur les sous-espaces de dimension p de ces cônes. Nous montrons alors un principe de contraction pour cette jauge. Cela nous permet de prouver, pour un opérateur contractant un tel cône, l'existence d'un trou spectral séparant les p valeurs propres dominantes du reste du spectre. Nous utilisons cette théorie pourdémontrer un théorème de régularité analytique pour les exposants de Lyapunov d'un produit aléatoire d'opérateurs contractant un même cône.Nous donnons également une comparaison entre la distance de Hausdorff entre espaces vectoriels et notre jauge.Enfin, nous introduisons une notion de cône dual pour les cônes p-dimensionnels. Dans ce cadre, nous prouvons que les propriétéstopologiques d'un cône se traduisent en propriétés topologiques sur son dual, et réciproquement. Nous complétons le théorème de régularitéprécédent en démontrant l'existence et la régularité d'une décomposition de l'espace en "espace lent" et "espace rapide".

The subject of this thesis is the introduction, the study and the applications of multidimensional complex cones. First, we study the grassmannian of Banach space. We define a notion of right decomposition for p-dimensional spaces and we prove the equivalence between theHausdorff distance on the grassmannian and the distance given by a norm on the exterior algebra.Then, we define p-dimensional complex cones and a gauge on the subspaces of dimension p of these cones. We show a contraction principle for thisgauge. This allows us to prove, for an operator contracting such a cone, the existence of a spectral gap which isolate the p leading eigenvaluesfrom the rest of the spectrum. We use this theory to prove a theorem of analytic regularity for Lyapunov exponents of a random product ofoperators contracting a cone. We also give a comparison between the Hausdorff distance for vector spaces and our gauge.Finally, we introduce a notion of dual cone for p-dimensional cones. In this setting, we prove that the topological properties of a cone translateinto topological properties for its dual and conversely. We complete the previous regularity theorem by proving the existence and the regularity ofa dominated splitting of the space into a "fast space" and a "slow space"., L'objet de cette thèse est l'introduction, l'étude et l'utilisation des cônes complexes multidimensionnels. Dans un premier temps, nous étudions la grassmannienne des espaces de Banach. Nous définissons une notion de bonne décomposition pour les espaces de dimension p et nous démontronsl'équivalence entre la distance de Hausdorff sur la grassmannienne et la distance fournie par une norme sur l'algèbre extérieure.Dans un deuxième temps, nous définissons les cônes complexes p-dimensionnels ainsi qu'une jauge sur les sous-espaces de dimension p de ces cônes. Nous montrons alors un principe de contraction pour cette jauge. Cela nous permet de prouver, pour un opérateur contractant un tel cône, l'existence d'un trou spectral séparant les p valeurs propres dominantes du reste du spectre. Nous utilisons cette théorie pourdémontrer un théorème de régularité analytique pour les exposants de Lyapunov d'un produit aléatoire d'opérateurs contractant un même cône.Nous donnons également une comparaison entre la distance de Hausdorff entre espaces vectoriels et notre jauge.Enfin, nous introduisons une notion de cône dual pour les cônes p-dimensionnels. Dans ce cadre, nous prouvons que les propriétéstopologiques d'un cône se traduisent en propriétés topologiques sur son dual, et réciproquement. Nous complétons le théorème de régularitéprécédent en démontrant l'existence et la régularité d'une décomposition de l'espace en "espace lent" et "espace rapide".

International audience; In the continuity of a recent paper ([6]), dealing with finite Markov chains, this paper proposes and analyzes a recursive algorithm for the approximation of the quasi-stationary distribution of a general Markov chain living on a compact metric space killed in finite time. The idea is to run the process until extinction and then to bring it back to life at a position randomly chosen according to the (possibly weighted) empirical occupation measure of its past positions. General conditions are given ensuring the convergence of this measure to the quasi-stationary distribution of the chain. We then apply this method to the numerical approximation of the quasi-stationary distribution of a diffusion process killed on the boundary of a compact set and to the estimation of the spectral gap of irreducible Markov processes. Finally, the sharpness of the assumptions is illustrated through the study of the algorithm in a non-irreducible setting.

Classically, finite modified logarithmic Sobolev inequalities are used to deduce a differential inequality for the evolution of the relative entropy with respect to the invariant measure. We will check that these inequalities are ill-behaved with respect, on one hand, to the symmetrization procedure, and on the other hand, to the umbrella sampling procedure for Poincaré inequalities. A short spectral proof of the latter method is given to estimate the spectral gap of a finite reversible Markov generator $L$ in terms of the spectral gap of the restrictions of $L$ on two subsets whose union is the whole state space and whose intersection is not empty.

In this dissertation, I study several rigidity properties of von Neumann algebras. In Chapter 1, we prove the relative solidity of Bernoulli crossed products of arbitrary type. This result is based on Popa's deformation/rigidity and generalizes a theorem of Chifan and Ioana in the tracial case. As a consequence, when the acting group is non-amenable, the crossed product is prime (cannot be decomposed nontrivially as a tensor product of two factors) and the associated equivalence relation is solid.In Chapter 2, we study full factors in relation with the spectral gap property. The main result is a spectral gap characterization of full type III factors which is similar to Connes' characterization in the tracial case. This allows us to better understand the structure of these factors and their automorphism group. We generalize a theorem of Jones by giving a sufficient condition for a crossed product to be full. This condition is necessary when the group is abelian. In particular, we obtain a complete characterization of the type III_1 whose core is full. In a joint work with C. Houdayer and P. Verraedt, we show that a tensor product of two full factors is also full and we compute its Connes invariants. We also prove a unique McDuff decomposition theorem that generalizes a result of Popa in the II_1 case. In Chapter 3, we study McDuff factors, i.e. those factors that can absorb tensorially the hyperfinite factor, as well as their counterpart in ergodic theory, the so-called stable equivalence relations. We obtain a new "spectral gap like" characterization of these properties, based on a maximality argument. With this refined characterization, we are able to prove the following rigidity result: a direct product of two stable equivalence relations is stable if and only if one of them is already stable. The analoguous problem on McDuff factors remains open, but we do give some partial results.; Dans cette thèse, je m'intéresse à diverses propriétés de rigidité des algèbres de von Neumann. Dans le Chapitre 1, je démontre la solidité relative des produits croisés issus d'actions Bernoulli de type quelconque. Ce résultat repose sur la théorie de la déformation/rigidité de Popa et généralise un théorème de Chifan et Ioana en type II. Comme conséquence, dès que le groupe qui agit est non-moyennable, ces produits croisés sont premiers (n'admettent pas de décomposition non triviale en produit tensoriel de deux facteurs) et la relation d'équivalence associée est solide. Le Chapitre 2 a pour thème les facteurs pleins et les phénomènes de trous spectraux. Je montre notamment que tout facteur plein de type III vérifie une propriété de trou spectral similaire à celle obtenue par Connes dans le cas II_1. Le trou spectral permet d'analyser plus finement la structure de ces facteurs et de leur groupe d'automorphismes. Je généralise ainsi un théorème de Jones en donnant une condition suffisante pour qu'un produit croisé soit plein. Cette condition est de plus nécessaire dans le cas où le groupe qui agit est abélien. Ceci permet de caractériser complètement les facteurs de type III_1 dont le cœur est plein. Dans un travail en collaboration avec C. Houdayer et P. Verraedt, nous montrons aussi qu'un produit tensoriel de deux facteurs pleins est encore plein et nous calculons ses invariants de Connes. Nous obtenons aussi un théorème d'unique décomposition McDuff qui généralise un résultat de Popa dans le cas II_1.Dans le Chapitre 3, je m'intéresse aux facteurs McDuff, i.e. qui ont la propriété d'absorber tensoriellement le facteur hyperfini, ainsi qu'à leur analogue en théorie ergodique, les relations d'équivalences stables. Je donne notamment une nouvelle caractérisation de cette propriété de stabilité qui repose sur un argument de maximalité. Cette caractérisation de type "trou spectral", plus fine que celle connue jusqu'alors, permet de démontrer le résultat de rigidité suivant: un produit direct de deux relations d'équivalences est stable si et seulement si l'une des deux est stable. Le problème similaire pour les facteurs McDuff reste ouvert, mais je donne quelques résultats partiels.

We propose a general edge index definition for two-dimensional Floquet topological phases based on a switch-function formalism. When the Floquet operator has a spectral gap the index covers both clean and disordered phases, anomalous or not, and does not require the bulk to be fully localized. It is interpreted as a non-adiabatic charge pumping that is quantized when the sample is placed next to an effective vacuum. This vacuum is gap-dependent and obtained from a Floquet Hamiltonian. The choice of a vacuum provides a simple and alternative gap-selection mechanism. Inspired by the model from Rudner et al. we then illustrate these concepts on Floquet disordered phases. Switch-function formalism is usually restricted to infinite samples in the thermodynamic limit. Here we circumvent this issue and propose a numerical implementation of the edge index that could be adapted to any bulk or edge index expressed in terms of switch functions, already existing for many topological phases., Comment: 13 pages, 10 figures, Correcting some typos and including an application to a non-anomalous model. To appear in Physical Review B

We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type., 29 pages, minor revisions

Consider a Markov chain $(X_n)_{n\geqslant 0}$ with values in the state space $\mathbb X$. Let $f$ be a real function on $\mathbb X$ and set $S_0=0,$ $S_n = f(X_1)+\cdots + f(X_n),$ $n\geqslant 1$. Let $\mathbb P_x$ be the probability measure generated by the Markov chain starting at $X_0=x$. For a starting point $y \in \mathbb R$ denote by $��_y$ the first moment when the Markov walk $(y+S_n)_{n\geqslant 1}$ becomes non-positive. Under the condition that $S_n$ has zero drift, we find the asymptotics of the probability $\mathbb P_x ( ��_y >n )$ and of the conditional law $\mathbb P_x ( y+S_n\leqslant \cdot\sqrt{n} | ��_y >n )$ as $n\to +\infty.$, Figure 1 corrected

International audience; We continue our investigation of the intertwining relations for Markov semigroups and extend the results of [9] to multi-dimensional diffusions. In particular these formulae entail new functional inequalities of Brascamp-Lieb type for log-concave distributions and beyond. Our results are illustrated by some classical and less classical examples.

This paper is devoted to [Formula: see text]-entropies applied to Fokker–Planck and kinetic Fokker–Planck equations in the whole space, with confinement. The so-called [Formula: see text]-entropies are Lyapunov functionals which typically interpolate between Gibbs entropies and [Formula: see text] estimates. We review some of their properties in the case of diffusion equations of Fokker–Planck type, give new and simplified proofs, and then adapt these methods to a kinetic Fokker–Planck equation acting on a phase space with positions and velocities. At kinetic level, since the diffusion only acts on the velocity variable, the transport operator plays an essential role in the relaxation process. Here we adopt the [Formula: see text] point of view and establish a sharp decay rate. Rather than giving general but quantitatively vague estimates, our goal here is to consider simple cases, benchmark available methods and obtain sharp estimates on a key example. Some [Formula: see text]-entropies give rise to improved entropy–entropy production inequalities and, as a consequence, to faster decay rates for entropy estimates of solutions to non-degenerate diffusion equations. We prove that faster entropy decay also holds at kinetic level away from equilibrium and that optimal decay rates are achieved only in asymptotic regimes.

Let X be a solenoid, that is, a compact finite dimensional connected abelian group with normalized Haar measure m, and let G be a countable discrete group acting on X by continuous affine transformations. We show that the probability measure preserving action of G on (X,m) does not have the spectral gap property if and only if there exists a p(G)-invariant proper subsolenoid Y of X such that the image of G in the affine group Aff(X/Y) of X/Y is a virtually solvable group, where p(G) is the automorphism part of G. When G is finitely generated or when X is a p-adic solenoid, the subsolenoid Y can be chosen so that the image of G in Aff(X/Y) is virtually abelian. In particular, an action of a group by affine transformations on a solenoid has the spectral gap property if and only if this action is strongly ergodic., Typos corrected; complete details given for proof of Theorem 5

We study the properties of stationary states associated with nonequilibrium dynamics from a theoretical and a numerical point of view. These dynamics are obtained by perturbing equilibrium dynamics with mechanical and / or thermal forcings. In the theoretical approach, the system considered evolves according to a Langevin dynamics perturbed by a torque. In this framework, we study the convergence of the law of dynamics to the stationary measure, giving quantitative estimates of the exponential rate, both in the Hamiltonian and `` overdamped '' regimes.By a numerical approach, we consider a chain of rotors subjected to both forcings and a chain of Toda oscillators subject to a thermal forcing and a stochastic perturbation. We study the features of the stationary state and analyze its transport properties. In particular, in the case of the rotor chain, contrary to what is naively expected, we observe that the average energy current is in some cases increased by an opposite temperature gradient.; Nous étudions les propriétés des états stationnaires et de dynamiques hors-équilibre, d’un point de vue théorique et numérique. Ces dynamiques sont obtenues en perturbant la dynamique d’équilibre par forçage mécanique et/ou thermique. Dans l’approche théorique, le système considéré évolue selon une dynamique de Langevin à laquelle on ajoute une force extérieure. Nous étudions la convergence de la loi de la dynamique vers la mesure stationnaire, en donnant des estimations quantitatives du taux, dans les régimes Hamiltonien et sur amorties. Dans l’approche numérique, nous considérons une chaîne de rotateurs soumise aux deux forçages et une chaîne d’oscillateurs de Toda soumise à un forçage thermique et à une perturbation stochastique. Nous étudions les caractéristiques de l’état stationnaire et les propriétés de transport. Dans le cas de la chaîne de rotateurs nous observons en particulier que le courant d’énergie moyen est dans certains cas accru par un gradient de température opposé.

We study the properties of stationary states associated with nonequilibrium dynamics from a theoretical and a numerical point of view. These dynamics are obtained by perturbing equilibrium dynamics with mechanical and / or thermal forcings. In the theoretical approach, the system considered evolves according to a Langevin dynamics perturbed by a torque. In this framework, we study the convergence of the law of dynamics to the stationary measure, giving quantitative estimates of the exponential rate, both in the Hamiltonian and `` overdamped '' regimes.By a numerical approach, we consider a chain of rotors subjected to both forcings and a chain of Toda oscillators subject to a thermal forcing and a stochastic perturbation. We study the features of the stationary state and analyze its transport properties. In particular, in the case of the rotor chain, contrary to what is naively expected, we observe that the average energy current is in some cases increased by an opposite temperature gradient.; Nous étudions les propriétés des états stationnaires et de dynamiques hors-équilibre, d’un point de vue théorique et numérique. Ces dynamiques sont obtenues en perturbant la dynamique d’équilibre par forçage mécanique et/ou thermique. Dans l’approche théorique, le système considéré évolue selon une dynamique de Langevin à laquelle on ajoute une force extérieure. Nous étudions la convergence de la loi de la dynamique vers la mesure stationnaire, en donnant des estimations quantitatives du taux, dans les régimes Hamiltonien et sur amorties. Dans l’approche numérique, nous considérons une chaîne de rotateurs soumise aux deux forçages et une chaîne d’oscillateurs de Toda soumise à un forçage thermique et à une perturbation stochastique. Nous étudions les caractéristiques de l’état stationnaire et les propriétés de transport. Dans le cas de la chaîne de rotateurs nous observons en particulier que le courant d’énergie moyen est dans certains cas accru par un gradient de température opposé.

A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs.

In application, we would like to generate random numbers with a precise law MCMC (Markov Chaine Monte Carlo). The method consists in finding a diffusion which has the desired invariant law and in showing the convergence of this diffusion towards its equilibrium with an exponential rate. The exponent of this convergence is the spectral gap of the generator. It was shown by C.-R. Hwang, S.-Y. Hwang-Ma and S.-J. Sheu that the spectral gap can grow up by adding a non-symmetric term to the self-adjoint generator.This corresponds to passing from a reversible diffusion to a non-reversible diffusion. A means of constructing a non-reversible diffusion with the same invariant measure is to add an incompressible flow to the dynamics of the reversible diffusion.In this thesis, we study the behavior of diffusion when the flow is accelerated by multiplying the field of the vectors which describes it by a large constant. In 2008, P. Constantin, A. Kisekev, L. Ryzhik and A. Zlatoˇs have shown that if the flow was weakly mixing then the acceleration of the flow was sufficient to converge the diffusion towards its equilibrium after finite time. In this work, the speed of this phenomenon is explained under a condition of correlation of the flow. The article by B. Franke, C.-R.Hwang, H.-M. Pai and S.-J.Sheu (2010) gives the asymptotic expression of the spectral gap when the large constant goes to infinity. Here we are also interested in the speed with which the phenomenon manifests itself. First, we study the special case of an Ornstein-Uhlenbeck diffusion which is perturbed by a flow preserving the Gaussian measure. In this case, thanks to a result of G. Metafune, D. Pallara and E. Priola (2002), we can reduce the study of the generator spectrum to eigenvalues of a family of matrices. We study this problem with methods of limited development of eigenvalues. This problem is solved explicitly in this thesis and we also give a boundary for the convergence radius of the development. We then generalize this method in the case of a general diffusion in a formal way. These results may be useful to have a first idea on the speeds of convergence of the spectral gap described in the article by Franke et al. (2010)., En application, on souhaite générer des nombres aléatoires avec une loi précise (méthode de Monte Carlo par chaines de Markov - MCMC (Markov Chaine Monte Carlo)). La méthode consiste à trouver une diffusion qui a la loi invariante souhaitée et à montrer la convergence de cette diffusion vers son équilibre avec une vitesse exponentielle. L’exposant de cette convergence est le trou spectral du générateur. Il a été montré par Chii-Ruey Hwang, Shu-Yin Hwang-Ma, et Shuenn-Jyi Sheu qu’on peut agrandir le trou spectral, en rajoutant un terme non-symétrique au générateur auto-adjoint (souvent utilisé en MCMC). Ceci correspond à passer d’une diffusion réversible (en detailed balance) à une diffusion non réversible. Un moyen de construire une diffusion non-réversible avec la même mesure invariante est de rajouter un flot incompressible à la dynamique de la diffusion réversible.Dans cette thèse, nous étudions le comportement de la diffusion lorsqu’on accélère le flot sous-jacent en multipliant le champ des vecteurs qui le décrit par une grande constante. P. Constantin, A.Kisekev, L.Ryzhik et A.Zlatoš (2008) ont montré que si le flot était faiblement mélangeant alors l’accélération du flot suffisait pour faire converger la diffusion vers son équilibre en un temps fini. Dans ce travail, on explicite la vitesse de ce phénomène sous une condition de corrélation du flot. L’article de B. Franke, C.-R.Hwang, H.-M. Pai et S.-J. Sheu (2010) donne l’expression asymptotique du trou spectral lorsque le flot sous-jacent est accéléré vers l’infini. Ici aussi, on s’intéresse à la vitesse avec laquelle le phénomène se manifeste. Dans un premier temps, nous étudions le cas particulier d’une diffusion du type Ornstein-Uhlenbeck qui est perturbée par un flot préservant la mesure gaussienne. Dans ce cas, grâce à un résultat de G. Metafune, D. Pallara et E. Priola (2002), nous pouvons réduire l’étude du spectre du générateur à des valeurs propres d’une famille de matrices. Nous étudions ce problème avec des méthodes de développement limité des valeurs propres. Ce problème est résolu explicitement dans cette thèse et nous donnons aussi une borne pour le rayon de convergence du développement. Nous généralisons ensuite cette méthode dans le cas d’une diffusion générale de façon formelle. Ces résultats peuvent être utiles pour avoir une première idée sur les vitesses de convergence du trou spectral décrites dans l’article de Franke et al. (2010).

We consider the weakly asymmetric exclusion process with $N=L/2$ particles on a periodic lattice of $L$ sites, and hopping rates $1$ and $q=1-\mu/\sqrt{L}$ respectively in the forward and in the backward direction. Using Bethe ansatz, we obtain a systematic perturbative expansion of the spectral gap near $\mu=0$ by solving order by order a simple functional equation. A key point is that when $\mu\to0$, Bethe roots at a distance $1/\sqrt{L}$ from the edge of the Fermi sea should not be considered as a continuum, but converge instead at large $L$ to the complex zeroes of $1+\mathrm{erf}(x)$ after a rescaling by $\sqrt{L}$., Comment: 28 pages, 5 figures

In this paper we study two semigroups of completely positive unital self-adjoint maps on the von Neumann algebras of the free orthogonal quantum group $O_N^+$ and the free permutation quantum group $S_N^+$. We show that these semigroups satisfy ultracontractivity and hypercontractivity estimates. We also give results regarding spectral gap and logarithmic Sobolev inequalities., 19 pages

DOI 10.1007/s00440-016-0719-z; International audience; Consider the product $G_{n}=g_{n} \dots g_{1}$ of the random matrices $g_{1},\dots,g_{n}$ in $GL\left( d,\mathbb{R}\right) $ and the random process $G_{n}v=g_{n}\dots g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \left\{ 0\right\} .$ It is well known that under appropriate assumptions, the sequence $\left( \log \left\Vert G_{n}v\right\Vert \right) _{n\geq 1}$ behaves like a sum of i.i.d.~r.v.'sand satisfies standard classical properties such as the law of largenumbers, law of iterated logarithm and the central limit theorem. Denote by $\mathbb{B}$ the closed unit ball in $\mathbb{R}^{d}$ and by $\mathbb{B}^{c}$its complement. For any $v\in \mathbb{B}^{c}$ define the exit time of therandom process $G_{n}v$ from $\mathbb{B}^{c}$ by $\tau _{v}=\min \left\{n\geq 1:G_{n}v\in \mathbb{B}\right\} .$ We establish the asymptotic as $n\rightarrow \infty $ of the probability of the event $\left\{ \tau_{v}>n\right\} $ and find the limit law for the quantity $\frac{1}{\sqrt{n}} \log \left\Vert G_{n}v\right\Vert $ conditioned that $\tau _{v}>n.$

International audience; In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces.

International audience; In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincaré inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.

We consider a general multivariate affine stochastic recursion and the associated Markov chain on $\mathbb R^{d}$. We assume a natural geometric condition which implies existence of an unbounded stationary solution and we show that the large values of the associated stationary process follow extreme value properties of classical type, with a non trivial extremal index. We develop some explicit consequences such as convergence to Fr{\'e}chet's law or to an exponential law, as well as convergence to a stable law. The proof is based on a spectral gap property for the action of associated positive operators on spaces of regular functions with slow growth, and on the clustering properties of large values in the recursion.