Abstract. We define a notion of a measured length space X having nonnegative N-Ricci curvature, for N ∈ [1, ∞), or having ∞-Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [10] and [41] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix G. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general

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We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with (+)-boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where there are multiple phases but the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hardcore constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard–core lattice gas (independent sets).

Abstract. — We analyze recent proofs of decay of correlations and logarithmic Sobolev inequalities for unbounded spin systems in the perturbative regime developed by B. Zegarlinski, N. Yoshida, B. Helffer, Th. Bodineau. We investigate to this task a simple analytic model. Proofs are short and self-contained. Let µ be a probability measure on R satisfying, for some constant C> 0 and for every smooth enough function f on R, either the Poincaré (or spectral gap) inequality Varµ(f) ≤ C f ′2 dµ where Varµ(f) is the variance of f with respect to µ (see below), or the logarithmic

Abstract. µ being a nonnegative measure satisfying some Log-Sobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some Log-Sobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied. Résumé. µ étant une mesure positive satisfaisant une inégalité de Sobolev logarithmique, nous donnons des conditions sur F pour que la mesure de Boltzmann ν = e −2F µ satisfasse également une telle inégalité (améliorant et complétant ainsi la dernière partie de [6]). Les conditions obtenues sont illustrées par des exemples.

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We provide a new exponential concentration inequality for First Passage Percolation valid for a wide class of edge times distributions. This improves and extends a result by Benjamini, Kalai and Schramm [5] which gave a variance bound for Bernoulli edge times. Our approach is based on some functional inequalities extending the work of Rossignol [20], Falik and Samorodnitsky [9]. Résumé: On obtient une nouvelle inégalité de concentration exponentielle pour la percolation de premier passage, valable pour une large classe de distributions des temps d’arêtes. Ceci améliore et étend un résultat de Benjamini, Kalai et Schramm [5] qui donnait une borne sur la variance pour des temps d’arêtes suivant une loi de Bernoulli. Notre approche se fonde sur des inégalités fonctionnelles étendant les travaux de Rossignol [20], Falik et Samorodnitsky [9].

We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem. 1

Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (Lyapunov-Poincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by Hérau-Nier, Helffer-Nier and Villani is in particular discussed in the final section.

Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1.