Entropies, convexity, and functional inequalities

Our aim is to provide a short and self contained synthesis which generalise and unify various related and unrelated works involving what we call Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies can be seen in particular as an inclusive interpolation between Poincare and Gross logarithmic Sobolev inequalities. In addition to the known material, extensions are provided and improvements are given for some aspects. Stability by tensor products, convolution, and bounded perturbations are addressed. We show that under simple convexity assumptions on Phi, such inequalities hold in a lot of situations, including hyper-contractive diffusions, uniformly strictly log-concave measures, Wiener measure (paths space of Brownian Motion on Riemannian Manifolds) and generic Poisson space (includes paths space of some pure jumps Levy processes and related infinitely divisible laws). Proofs are simple and relies essentially on convexity. We end up by a short parallel inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic Gases and Information Theories.Comment: Formerly "On Phi-entropies and Phi-Sobolev inequalities". Author's www homepage: http://www.lsp.ups-tlse.fr/Chafai

Circular law for non-central random matrices

Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let \la_{n,1},...,\la_{n,n} be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution \frac{1}{n}(\de_{\la_{n,1}}+...+\de_{\la_{n,n}}) converges weakly as $n\to\infty$ to the uniform law over the unit disc \{z\in\dC;|z|\leq1\}. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\leq j,k\leq n}$ provided that $\mathrm{Tr}(MM^*)=O(n^2)$ and \mathrm{rank}(M)=O(n^\al) with \al<1. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur theorems.Comment: accepted in Journal of Theoretical Probabilit

Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems

Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau [LPY] on spectral gap and logarithmic Sobolev inequalities for unbounded conservative spin systems, we study uniform bounds in these inequalities for Glauber dynamics of Hamiltonian of the form V(x_1) + ... + V(x_n) + V(M-x_1 -...-x_n), (x_1,...,x_n) in R^n Specifically, we examine the case V is strictly convex (or small perturbation of strictly convex) and, following [LPY], the case V is a bounded perturbation of a quadratic potential. By a simple path counting argument for the standard random walk, uniform bounds for the Glauber dynamics yields, in a transparent way, the classical L^{-2} decay for the Kawasaki dynamics on d-dimensional cubes of length L. The arguments of proofs however closely follow and make heavy use of the conservative approach and estimates of [LPY], relying in particular on the Lu-Yau martingale decomposition and clever partitionings of the conditional measure.Comment: 20 pages. Accepted for publication in ``Markov Processes and Related Fields'

Around the circular law

These expository notes are centered around the circular law theorem, which 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. We present a proof in a Gaussian case, due to 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. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.Comment: Added: one reference and few comment

On the strong consistency of asymptotic M-estimators

The aim of this article is to simplify Pfanzagl's proof of consistency for asymptotic maximum likelihood estimators, and to extend it to more general asymptotic M-estimators. The method relies on the existence of a sort of contraction of the parameter space which admits the true parameter as a fixed point. The proofs are short and elementary.Comment: Accepted for publication in Journal of Statistical Planning and Inferenc

Explicit formulas for a continuous stochastic maturation model. Application to anticancer drug pharmacokinetics/pharmacodynamics

We present a continuous time model of maturation and survival, obtained as the limit of a compartmental evolution model when the number of compartments tends to infinity. We establish in particular an explicit formula for the law of the system output under inhomogeneous killing and when the input follows a time-inhomogeneous Poisson process. This approach allows the discussion of identifiability issues which are of difficult access for finite compartmental models. The article ends up with an example of application for anticancer drug pharmacokinetics/pharmacodynamics.Comment: Revised version, accepted for publication in Stochastic Models (Taylor & Francis

Contributions à l'étude de modèles biologiques, d'inégalités fonctionnelles, et de matrices aléatoires

The presented works concern three autonomous topics :(1) Biological models and Statistics : comparmental models, population pharmacokinetics and pharmacodynamics, estimators for stochastic inverse problems, nonlinear mixed effects models, mixture models, EM and ICF type algorithms, graphical covariance models, modelling in Cancerology, point processes and particles, queueing systems, Feynman-Kac formulas(2) Functional inequalities: Sobolev type inequalities, concentration of measure, isoperimetry, role of convexity in entropic inequalities, tensorization, heat kernels, Heisenberg group and hypoelliptic dynamics, queueing systems, mixtures of distributions (3) Random matrices: spectrum of random Markov matrices, graphs with random weights, Wigner, Marchenko-Pastur, and Girko-Bai type theorems, convergence of extremal eigenvalues, rank one deformations.The most frequent concept here is the notion of Markov dynamics. In the first part, the compartmental models give rise to such dynamics. The second part is related to the geometry and trend to equilibrium of Markov dynamics. The third part is devoted to random Markov dynamics. However, these three parts cannot be reduced to the study of certain aspects of Markov dynamics. The content ranges from concrete applications to abstract theoretical problems, and makes use of various concepts and techniques from Mathematical Analysis, Probability Theory and Statistics.Les travaux présentés concernent trois thématiques autonomes :(1) Modèles biologiques et statistique : modèles compartimentaux, pharmacocinétique et pharmacodynamie de population, estimateurs pour problèmes inverses stochastiques, modèles non-linéaires à effets mixtes, modèles de mélanges, algorithmes de type EM et ICF, modèles graphiques de covariance, modélisation en cancérologie, processus ponctuels, particules, files d'attentes, renormalisation de processus markoviens inhomogènes et formules de Feynman-Kac(2) Inégalités fonctionnelles : inégalités de type Sobolev, concentration de la mesure, isopérimétrie rôle de la convexité dans les inégalités entropiques, tensorisation, noyau de la chaleur, groupe d'Heisenberg et dynamiques hypoelliptiques, files d'attentes, mélanges de lois (3) Matrices aléatoires : spectre des matrices markoviennes aléatoires, graphes à poids aléatoires, théorèmes de type Wigner, Marchenko-Pastur, et Girko-Bai, convergence des valeurs propres extrémales, déformations de rang un.Le concept le plus récurrent ici est celui de dynamique markovienne. Dans la première partie, ce sont les modèles à compartiments de la pharmacologie qui sont liés à de telles dynamiques. La seconde partie traite d'inégalités fonctionnelles associées à la vitesse et à la géométrie de dynamiques markoviennes. Enfin, la troisième partie traite de dynamiques markoviennes aléatoires. Ces trois parties ne se réduisent pas à l'étude de facettes de problèmes markoviens. Leur contenu balaye un spectre à la fois théorique et appliqué, et met en oeuvre des techniques et des concepts variés issus de l'analyse, des probabilités, et de la statistique

Dynamics of a planar Coulomb gas

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 Poincar{\'e} 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.Comment: Minor revision for Annals of Applied Probabilit

Spectrum of large random reversible Markov chains: two examples

We take on a Random Matrix theory viewpoint to study the spectrum of certain reversible Markov chains in random environment. As the number of states tends to infinity, we consider the global behavior of the spectrum, and the local behavior at the edge, including the so called spectral gap. Results are obtained for two simple models with distinct limiting features. The first model is built on the complete graph while the second is a birth-and-death dynamics. Both models give rise to random matrices with non independent entries.Comment: accepted in ALEA, March 201