Large deviations, a phase transition, and logarithmic Sobolev inequalities in the block spin Potts model

Modified logarithmic Sobolev inequalities and transportation inequalities

Logarithmic Sobolev Inequalities for Unbounded Spin Systems Revisited

where Entμ(f ) is the entropy of f with respect to μ (see below). It is well-known that the product measure μ of μ on R then satisfies the preceding inequalities (with the Euclidean length of the gradient of the function f on R) with the same constant C, in particular independent of the dimension n. Let now H be a smooth function on R such that ∫ edμ < ∞. Define Q the probability measure on R w...

Logarithmic Sobolev Inequalities and the Information Theory

In this paper we present an overview on logarithmic Sobolev inequalities. These inequalities have become a subject of intense research activity during the past years, from analysis and geometry in finite and infinite dimension, to probability and statistical mechanics, and of course many others developments and applications are expected. We have divided this paper into three parts. The first pa...

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for d...

Modified Logarithmic Sobolev Inequalities in Discrete Settings

Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare’ (spectral gap) inequality. We sho...