. We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial dierential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information. Contents 1. Introduction 1 2. Main results 5 3. Heuristics 11 4. Proof of Theorem 1 18 5. Proof of Theorem 3 24 6. An application of Theorem 1 30 7. Linearizations 31 Appendix A. A nonlinear approximation argument 35 References 36 1. Introduction Let M be a smooth complete Riemannian manifold of dimension n, with the geodesic distance d(x; y) = inf 8 < : s Z 1 0 j _ w(t)j 2 dt; w 2 C 1 ((0; 1); M); w(0) = x; w(1) = y 9 ...

Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications. 1.

We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to non-Gaussian Wigner and Wishart matrices. We also provide concentration bounds for non commutative functionals of random matrices. 1 Introduction and statement of results Consider a random N N Hermitian matrix X with i.i.d. complex entries (except for the symmetry constraint) satisfying a moment condition. It is well known since Wigner [28] that the spectral measure of N 1=2 X converges to the semicircle law. This observation has been generalized to a large class of matrices, e.g. sample covariance matrices of the form XRX where R is a deterministic diagonal matrix ([19]), band matrices (see [5, 16, 20]), etc. For the Wigner case, this convergence has been supplemented by Central Limit Theorems, see [15] for the case of Gaussian entries and [17], [22] for the gen...

this paper is to provide such general-purpose inequalities. Our approach is based on a generalization of Ledoux's entropy method (see [26, 28]). Ledoux's method relies on abstract functional inequalities known as logarithmic Sobolev inequalities and provide a powerful tool for deriving exponential inequalities for functions of independent random variables, see Boucheron, Massart, and AMS 1991 subject classifications. Primary 60E15, 60C05, 28A35; Secondary 05C80 Key words and phrases. Moment inequalities, Concentration inequalities; Empirical processes; Random graphs Supported by EU Working Group RAND-APX, binational PROCOPE Grant 05923XL The work of the third author was supported by the Spanish Ministry of Science and Technology and FEDER, grant BMF2003-03324 Lugosi [6, 7], Bousquet [8], Devroye [14], Massart [30, 31], Rio [36] for various applications. To derive moment inequalities for general functions of independent random variables, we elaborate on the pioneering work of Latala and Oleszkiewicz [25] and describe so-called #-Sobolev inequalities which interpolate between Poincare's inequality and logarithmic Sobolev inequalities (see also Beckner [4] and Bobkov's arguments in [26])

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. We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenters are such that the Gibbs specication satises a certain mixing condition, then the spectral gap of the generator is strictly positive uniformly in the volume and boundary condition. The required mixing condition holds if, for instance, there is a convergent cluster expansion. Key Words: Spectral gap, Gibbs measures, continuous systems, birth and death processes Mathematics Subject Classication: 82C21, 60K35, 82C22, 60J75 This work was partially supported by GNAFA and by \Conanziamento Murst" v1.4 1. Introduction We consider a continuous gas in a bounded volume R d , distributed according the Gibbs probability measure associated to a nite range pair potential '. The Gibbs measu...

. ---We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals, such as the graph distance, in this setting. 1. Introduction The classical logarithmic Sobolev inequality for Brownian motion B = (B t ) t0 in IR d [Gr] indicates that for all functionals F in the domain of the Malliavin gradient operator D : L 2(\Omega ; IP) ! L 2(\Omega \Theta [0; T ]; IP\Omega dt), IE(F 2 log F 2 ) \Gamma IE(F 2 ) log IE(F 2 ) 2 IE `Z T 0 jD t F j 2 dt ' : (1:1) In particular, if F = f(B t 1 ; : : : ; B t n ), 0 = t 0 t 1 \Delta \Delta \Delta t n for some smooth function f : (IR d ) n ! IR, D t F = n X i=1 r i F I ft...

. We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several dierent techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way. Key Words: Entropy, Logarithmic Sobolev inequalities, Gibbs measures Mathematics Subject Classication 2000: 82B20, 82C20, 39B62 v1.0 1. Introduction Logarithmic Sobolev inequalities have been introduced in [Gr1] where it has been shown that Z R d f 2 (x) log jf(x)j d (dx) Z R d jr...

Abstract. Concentration inequalities deal with deviations of functions of independent random variables from their expectation. In the last decade new tools have been introduced making it possible to establish simple and powerful inequalities. These inequalities are at the heart of the mathematical analysis of various problems in machine learning and made it possible to derive new efficient algorithms. This text attempts to summarize some of the basic tools. 1

. Let (Z t ) t2R+ be a martingale in L 4 having the chaos representation property and angle bracket dhZ t ; Z t i = dt. We show that the positive functionals F of (Z t ) t2R+ satisfy the modied logarithmic Sobolev inequality E[F log F ] E[F ] log E[F ] 1 2 E 1 F Z 1 0 (2 i t )(D t F ) 2 dt ; where D is the gradient operator dened by lowering the degree of multiple stochastic integrals with respect to (Z t ) t2R+ and (i t ) t2R+ f0; 1g is a process given by the structure equation satised by (Z t ) t2R+ . Resume. Soit (Z t ) t2R+ une martingale dans L 4 qui satisfait la propriete de representation chaotique, avec dhZ t ; Z t i = dt. On montre que les fonctionnelles positives F de (Z t ) t2R+ satisfont l'inegalite de Sobolev logarithmique modiee E[F log F ] E[F ] log E[F ] 1 2 E 1 F Z 1 0 (2 i t )(D t F ) 2 dt ; ou D est l'operateur gradient qui abaisse le degre des integrales stochastiques multiples par rapport a (Z t ) t2R+ , et (i t ) t2R+ f0; 1g est un processus donne par l'equation de structure satisfaite par (Z t ) t2R+ . Key words: Logarithmic Sobolev inequalities, normal martingales, Azema martingales, Poisson random measures. Mathematics Subject Classication. 60G44, 60G60, 46E35, 46E39. 1