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Paul-Marie Samson

Publications and source records attributed to Paul-Marie Samson.

16 recordsLinked to original sources
arXiv2023

Criteria for entropic curvature on graph spaces

In this paper we establish new simple local geometric criteria for discrete entropic curvature introduced in [47] that are powerful enough to capture many geometric properties of complex models arising in mathematical physics. These results are robust in the sense that they apply to any discrete graph equipped with a Markov reversible generator. Our definitions of entropic curvature differ from the one of the pioneering works of Erbar-Maas [19,20] (which is already a discrete analog of the Lott-Sturm-Villani entropic curvature in the continuous setting). Singularly, our results provide refined concentration properties related to the celebrated convex-hull method by Talagrand [50,51] for a large class of probability measures that cannot be captured from Erbar- Maas entropic definition of curvature. Our approach gives also a new insight of the convex hull method, without being related to induction arguments. We illustrate the power of our results, as well as the general entropic strategy developed in this paper, to tackle challenging models studied in mathematical physics, including Gibbs measures with interaction potentials such as Ising models on the discrete hypercube and measures with interaction potential on the lattice $\mathbb{Z}^n$. For instance, we significantly improve the constant of the refined convex concentration properties obtained in [2] for Ising models. Moreover, when dealing with the antiferromagnetic Curie-Weiss model, we improve the previously known bound for entropic curvature by a factor of $\sqrt{n}$. Our simple criteria also provides the expected right order of magnitude $C/\sqrt n$ for the lower-bound on the entropic curvature for the renowned Sherrington-Kirkpatrick model from the spin glass theory. This last result is consistant with the recent works [6,17] on the modified logarithmic Sobolev and Poincar\'e inequalities for the Sherrington-Kirkpatrick model.

math.PR↗
arXiv2022

Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $L^1$

It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in $L^p$ for $p>1$. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in $\mathbb{R}^n$, as well as for the $M/M/\infty$ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.

math.PR↗
arXiv2022

Entropic curvature on graphs along Schr{ö}dinger bridges at zero temperature

Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{é}onard by replacing W2-Wasserstein geodesics by Schr{ö}odinger bridges in the definition of entropic curvature [23, 25, 24]. As a remarkable fact, as a temperature parameter goes to zero, these Schr{ö}dinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice Z n endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar-Maas results on graphs [27, 10, 11], entropic curvature results of this paper imply new Pr{é}kopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} n and for the Bernoulli Laplace model, a new W2 -- W1 transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see [28, 15]) associated to the so-called convex-hull method by Talagrand [38].

math.PR↗
arXiv2019

Transport Proofs Of Some Discrete Variants Of The Pr{é}Kopa-leindler Inequality

We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Pr{é}kopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].

math.PR↗
arXiv2017

Transport-entropy inequalities on locally acting groups of permutations

Following Talagrand's concentration results for permutations picked uniformly at random from a symmetric group [Tal95], Luczak and McDiarmid have generalized it to more general groups G of permutations which act suitably 'locally'. Here we extend their results by setting transport-entropy inequalities on these permutations groups. Talagrand and Luczak-Mc-Diarmid concentra- tion properties are consequences of these inequalities. The results are also gen- eralised to a larger class of measures including Ewens distributions of arbitrary parameter $θ$ on the symmetric group. By projection, we derive transport-entropy inequalities for the uniform law on the slice of the discrete hypercube and more generally for the multinomial law. These results are new examples, in discrete setting, of weak transport-entropy inequalities introduced in [GRST15], that con- tribute to a better understanding of the concentration properties of measures on permutations groups. One typical application is deviation bounds for the so- called configuration functions, such as the number of cycles of given lenght in the cycle decomposition of a random permutation.

math.PR↗
arXiv2017

Deviation inequalities for convex functions motivated by the Talagrand conjecture

Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.

math.PR↗
arXiv2015

Kantorovich duality for general transport costs and applications

We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.

math.PR↗
arXiv2015

Characterization of a class of weak transport-entropy inequalities on the line

We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities in dimension one. In particular, we obtain a weak transport-entropy form of the convex Poincar{é} inequality in dimension one.

math.PR↗
arXiv2012

Displacement convexity of entropy and related inequalities on graphs

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.

math.PR↗
arXiv2012

Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

We prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton- Jacobi equations on a general metric space. As a first consequence, we show in full gener- ality that the log-Sobolev inequality is equivalent to an hypercontractivity property of the Hamilton-Jacobi semi-group. As a second consequence, we prove that Talagrand's transport- entropy inequalities in metric space are characterized in terms of log-Sobolev inequalities restricted to the class of c-convex functions.

math.PR↗