CHAFAI Djalil

Coulomb gases are special probability distributions, related to potential theory, that appear at many places in pure and applied mathematics and physics. In these short expository notes, we focus on some models, ideas, and structures. We present briefly selected mathematical aspects, mostly related to exact solvability, and to first and second order global asymptotics. A particular attention is devoted to two-dimensional exactly solvable models of random matrix theory such as the Ginibre model. Thematically, these notes lie between probability theory, mathematical analysis, and statistical physics, and aim to be very accessible. They form a contribution to a volume of the "Panoramas et Synthèses" series around the workshop "États de la recherche en mécanique statistique", organized by Société Mathématique de France, held at Institut Henri Poincaré, Paris, in the fall of 2018 (https://statmech2018.sciencesconf.org).

We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, entropy and Wasserstein. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.

The aim of this note is to provide a quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz $s$-kernels, including the logarithmic kernel, in arbitrary dimensions. The equilibrium measure is a radial arcsine distribution. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functions. These identities are not found in the existing tables for series and integrals, and are not recognized by advanced mathematical software. Among other ingredients, our proofs involve the Euler-Lagrange variational characterization, the Funk-Hecke formula, and the Weyl lemma for the regularity of elliptic equations.

This book brings together a series of short articles written by researchers from the University of Paris Dauphine - PSL, and their co-authors, during the summer of 2020. In the context of their research work, and of a multidisciplinary working group created in March 2020, they have examined these questions from the perspective of their discipline: economics, management, sociology and political science, law, mathematics and computer science. This collection contributes to the analysis of the crisis and the responses that have been made to date, to what it has revealed about our societies. It underlines the complexity of the crisis and the importance of mobilizing multidisciplinary teams on multifaceted societal issues.

This note, mostly expository, is devoted to Poincaré and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.

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Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in probability. This result can also be seen as the convergence of the support in the circular law theorem under optimal moment conditions. In the proof we establish the convergence in law of the reciprocal characteristic polynomial to a random analytic function outside the unit disc, related to a hyperbolic Gaussian analytic function. The proof is short and differs from the usual approaches for the spectral radius. It relies on a tightness argument and a joint central limit phenomenon for traces of fixed powers.

We consider a one-dimensional classical Wigner jellium, not necessarily charge neutral, for which the electrons are allowed to exist beyond the support of the background charge. The model can be seen as a one-dimensional Coulomb gas in which the external field is generated by a smeared background on an interval. It is a true one-dimensional Coulomb gas and not a one-dimensional log-gas. We first observe that the system exists if and only if the total background charge is greater than the number of electrons minus one. Moreover we obtain a R\'enyi-type probabilistic representation for the order statistics of the particle system beyond the support of the background. Furthermore, for various backgrounds, we show convergence to point processes, at the edge of the support of the background. In particular, this provides asymptotic analysis of the fluctuations of the right-most particle. Our analysis reveals that these fluctuations are not universal, in the sense that depending on the background, the tails range anywhere from exponential to Gaussian-like behavior, including for instance Tracy-Widom-like behavior.

This note, mostly expository, is devoted to Poincaré and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.

We consider a planar Coulomb gas in which the external potential is generated by a smeared uniform background of opposite-sign charge on a disc. This model can be seen as a two-dimensional Wigner jellium, not necessarily charge neutral, and with particles allowed to exist beyond the support of the smeared charge. The full space integrability condition requires low enough temperature or high enough total smeared charge. This condition does not allow at the same time, total charge neutrality and determinantal structure. The model shares similarities with both the complex Ginibre ensemble and the Forrester--Krishnapur spherical ensemble of random matrix theory. In particular, for a certain regime of temperature and total charge, the equilibrium measure is uniform on a disc as in the Ginibre ensemble, while the modulus of the farthest particle has heavy-tailed fluctuations as in the Forrester--Krishnapur spherical ensemble. We also touch on a higher temperature regime producing a crossover equilibrium measure, as well as a transition to Gumbel edge fluctuations. More results in the same spirit on edge fluctuations are explored by the second author together with Raphael Butez.

These lecture notes cover the notions of probability in the program of the internal mathematics aggregation. They do not constitute models of oral lessons.

We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an I-continuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system.

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We study the long-time behavior of the dynamics of interacting planar Brow-nian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann – Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincaré inequalities are available. Moreover the second moment dynamics turns out to be a nice Cox – Ingersoll – Ross process in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.

Coulomb and log-gases are exchangeable singular Boltzmann-Gibbs measures appearing in mathematical physics at many places, in particular in random matrix theory. We explore experimentally an efficient numerical method for simulating such gases. It is an instance of the Hybrid or Hamiltonian Monte Carlo algorithm, in other words a Metropolis-Hastings algorithm with proposals produced by a kinetic or underdamped Langevin dynamics. This algorithm has excellent numerical behavior despite the singular interaction, in particular when the number of particles gets large. It is more efficient than the well known overdamped version previously used for such problems.

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Maïda and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty.

Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance, in other words that there are no outliers in the circular law. In this work we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.

These lecture notes cover the notions of probability in the program of the internal mathematics aggregation. They do not constitute models of oral lessons. The electronic version is available for free on HAL while the paper version is sold at cost on Amazon Europe.

The main object of this thesis is the study of several models of random polynomials. The aim is to understand the macroscopic behavior of random polynomial roots whose degree tends to infinity. We will explore the connection between the roots of random polynomials and Coulomb gases in order to obtain large deviation principles for the empirical measurement of the roots. We revisit the paper of Zeitouni and Zelditch which establishes a large deviation principle for a general model of random polynomials with complex Gaussian coefficients. We extend this result to the case of real Gaussian coefficients. Then, we show that these results remain valid for a large class of laws on coefficients, making large deviations a universal phenomenon for these models. Moreover, we prove all the previous results for the model of renormalized Weyl polynomials. We are also interested in the behavior of the root of largest modulus of Kac polynomials. This one has a non-universal behavior and is in general a random variable with heavy tails. Finally, we prove a principle of large deviations for the empirical measurement of biorthogonal sets.

We consider the random Markov matrix obtained by assigning i.i.d. non-negative weights to each edge of the complete oriented graph. In this study, the weights have unbounded first moment and belong to the domain of attraction of an alpha-stable law. We prove that as the dimension tends to infinity, the empirical measure of the singular values tends to a probability measure which depends only on alpha, characterized as the expected value of the spectral measure at the root of a weighted random tree. The latter is a generalized two-stage version of the Poisson weighted infinite tree (PWIT) introduced by David Aldous. Under an additional smoothness assumption, we show that the empirical measure of the eigenvalues tends to a non-degenerate isotropic probability measure depending only on alpha and supported on the unit disc of the complex plane. We conjecture that the limiting support is actually formed by a strictly smaller disc.

Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries.

This collection draws its source from the courses of master of applied mathematics and preparation to the test of modeling of the aggregation of mathematics. This book focuses on models rather than on tools, and each chapter is devoted to a model. The first target audience is teachers-researchers in probability, both beginners and experienced. Many chapters can also be of direct benefit to master's students or those preparing for the agrégation.

Presentation of the laureates' thesis work by a "senior" researcher, then by the laureate. The topics covered are diverse: the snows of the Antarctic plateau (climate physics), the mechanics of the cell (biology), the fight against infectious diseases (genetics), the brain in the man-machine interface (computer science) and randomness (mathematics).

This collection draws its source from the courses of master of applied mathematics and preparation to the test of modeling of the aggregation of mathematics. This book focuses on models rather than on tools, and each chapter is devoted to a model. The first target audience is teachers-researchers in probability, both beginners and experienced. Many chapters can also be of direct benefit to master's students or those preparing for the agrégation.

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two.

The Expectation-Maximization (EM) algorithm is one of the most important algorithms in statistics.

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Queues 1 are among the most common and useful random models. The simplest case to describe is probably the following: customers are queuing in front of a counter called server.

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his collection draws its source from our Master's courses in applied mathematics and from our preparation for the modeling test of the agrégation de mathématiques. It is a collection accumulated over time that we wish to share with pleasure and enthusiasm, designed to be consulted occasionally, at random, by leafing through the pages, consulting the table of contents or the index. The first target audience is teachers-researchers in probability, beginners or confirmed. We hope that this book will inspire them in the design of their master's courses or in the supervision of research training courses. Many chapters can also be of direct benefit to master's students or those preparing for the agrégation. The author's approach is to focus on models rather than tools, and to devote each chapter to a model. Although sometimes linked, the chapters are essentially autonomous and contain little or no reminders of the course.

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A headhunting firm wants to build a strong team thanks to a recruitment process: each candidate gets a score after his interview and the recruiters decide in the moment to hire him or not.

The second principle of thermodynamics postulates that for any isolated system, an extensive macroscopic quantity called entropy increases with time.

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We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

These expository notes propose to follow, across fields, some aspects of the concept of entropy. Starting from the work of Boltzmann in the kinetic theory of gases, various universes are visited, including Markov processes and their Helmholtz free energy, the Shannon monotonicity problem in the central limit theorem, the Voiculescu free probability theory and the free central limit theorem, random walks on regular trees, the circular law for the complex Ginibre ensemble of random matrices, and finally the asymptotic analysis of mean-field particle systems in arbitrary dimension, confined by an external field and experiencing singular pair repulsion. The text is written in an informal style driven by energy and entropy. It aims to be recreative and to provide to the curious readers entry points in the literature, and connections across boundaries.

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on Hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest.

We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

The circular law theorem states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. In these expository notes, we present a proof in a Gaussian case, due to Mehta and Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. We also discuss some related models and open problems.

We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan and Rider, boils down to identities in law and to an instance of the Laplace method.

We study a physical system of $N$ interacting particles in $\mathbb{R}^d$, $d\geq1$, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as $N$ tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension $d>2$, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as $N$ tends to infinity. In the more specific case of Coulomb interaction in dimension $d>2$, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.

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We study a random sequence with values in the permutations of finite sets, called the Chinese restaurant process. This process is related to Ewens' law, well known in elementary combinatorics. This process and this law are in some way an analogue for permutations of the Poisson process and the Poisson law, more classical in probability theory.

The texts in this volume present several aspects of the mathematics of randomness and highlight the many fields of application in which they are used. Sylvie Méléard uses population dynamics modeling to introduce Markovian jump processes. She highlights, through the study of large population approximations of the logistic birth and death process, an ordinary differential equation model and a stochastic differential equation model, depending on the time scale and population size considered. Christophe Giraud provides an introduction to the mathematical foundations of automatic classification, a theory that is used both in the filtering of email spam and in the automatic search for active molecules in medicine. His text presents various mathematical techniques used in the estimation of the probability of classification error by an algorithm. Djalil Chafaï develops some aspects of the theory of random matrices, which is a field of mathematics at the intersection of probability theory and linear algebra. This theory has applications in applied fields as well as in the most fundamental parts of mathematics.

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Given a birth-death process on N with semigroup (P_t) and a discrete gradient d_u depending on a positive weight u, we establish intertwining relations of the form d_u P_t = Q_t d_u, where (Q_t) is the Feynman-Kac semigroup with potential V_u of another birth-death process. We provide applications when V_u is positive and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity.

We investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if k<>j and L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős-Rényi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.

This thesis is centered around an algorithm for the construction of probability measures with prescribed marginal laws, called Iterative Proportional Fitting (IPF). Coming from statistics, this algorithm is based on successive projections on probability spaces with the Kullback-Leibler pseudo relative entropy distance. This thesis constitutes an overview of the available results on the subject, and contains some extensions and refinements. The first part is devoted to the study of relative entropy projections, to existence and uniqueness criteria as well as to characterization criteria related to the closure of a sum of subspaces. Under certain conditions, the problem becomes a maximum entropy problem for graphical marginal constraints. The second part highlights the iterative IPF process. Originally addressing an estimation problem for contingency tables, it is more generally an analogue of a classical algorithm of alternating projections on Hilbert spaces. After presenting the properties of the IPF, we focus on convergence results in the discrete finite case and in the Gaussian case, as well as in the continuous case with two marginals, for which an extension is proposed. We then focus on the Gaussian case, for which a new formulation of the IPF allows us to obtain a speed of convergence in the case with two prescribed marginals, whose optimality in dimension 2 is shown.

This thesis is part of the theory of random matrices, at the intersection with the theory of free probabilities and operator algebras. It is part of a general approach that has proven itself in the last decades: importing techniques and concepts from non-commutative probability theory for the study of the spectrum of large random matrices. We are interested here in generalizations of the asymptotic freedom theorem of Voiculescu. In Chapters 1 and 2, we show strong asymptotic freedom results for Gaussian, unitary random and deterministic matrices. In Chapters 3 and 4, we introduce the notion of false asymptotic freedom for deterministic matrices and some Hermitian matrices with independent subdiagonal entries, interpolating the Wigner and Lévy matrix models.

The thesis studies non-parametric methods (NP) for estimating the distribution of random effects in a non-linear mixed effects model. The objective is to evaluate the interest of these methods for population Pharmacokinetic (PK) and/or Pharmacodynamic (PD) analyses in the Pharmaceutical industry. First, the thesis reviews the statistical properties of four important NP methods. In addition, it evaluates their practical performance through simulation studies inspired by population PK analyses. The value of NP methods is established, both in theory and in practice. The NP methods are then evaluated for population PK/PD analysis of an antidiabetic drug. The objective is to evaluate the ability of the methods to detect a subpopulation of non-responders to the treatment. Simulation studies show that two NP methods seem to be better able to detect this subpopulation. The last part of the thesis is devoted to the research of stochastic algorithms to improve the computation of NP methods. A perturbed stochastic gradient algorithm is proposed.

The work presented concerns three autonomous themes: (1) Biological and statistical models: compartmental models, population pharmacokinetics and pharmacodynamics, estimators for stochastic inverse problems, nonlinear mixed-effects models, mixture models, EM and ICF algorithms, graphical covariance models, cancer modeling, point processes, particles, queues, renormalization of inhomogeneous Markov processes and Feynman-Kac formulas (2) Functional inequalities: Sobolev-type inequalities, measure concentration, isoperimetry role of convexity in entropic inequalities, tensorization, heat kernel, Heisenberg group and hypoelliptic dynamics, queues, mixtures of laws (3) Random matrices: spectrum of random Markovian matrices, random weight graphs, Wigner, Marchenko-Pastur, and Girko-Bai type theorems, convergence of extremal eigenvalues, rank-one deformations. The most recurrent concept here is that of Markovian dynamics. In the first part, compartmental models of pharmacology are related to such dynamics. The second part deals with functional inequalities associated with the speed and geometry of Markov dynamics. Finally, the third part deals with random Markov dynamics. These three parts are not reduced to the study of facets of Markovian problems. Their content covers a spectrum that is both theoretical and applied, and uses various techniques and concepts from analysis, probability, and statistics.

1°) Use of Bobkov functional inequalities for the establishment of quasi-Gaussian large deviation principles. 2°) Study of the logarithmic Sobolev inequality in information theory. 3°) Establishment of logarithmic Poincaré and Sobolev inequalities for some Kawasaki and Glauber dynamics for a continuous spin model in statistical mechanics.