Uniform Logarithmic Sobolev Inequalities Research Articles

Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems

In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincaré inequality is based on the work of Ledoux (In Séminaire de Probabilités, XXXV (2001) 167–194, Springer) and the uniform logarithmic Sobolev inequality is based on Zegarlinski’s theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from (J. Funct. Anal. 257 (2009) 4015–4033). The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKean–Vlasov equation with an explicit rate, We need here weaker conditions than the results of (Rev. Mat. Iberoam. 19 (2003) 971–1018) (by means of the displacement convexity approach), (Stochastic Process. Appl. 95 (2001) 109–132; Ann. Appl. Probab. 13 (2003) 540–560) (by Bakry–Emery’s technique) or the recent work (Arch. Ration. Mech. Anal. 208 (2013) 429–445) (by dissipation of the Wasserstein distance).

Uniform LSI for the canonical ensemble on the 1D-lattice with strong, finite-range interaction

We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show that the canonical ensemble (ce) satisfies a uniform logarithmic Sobolev inequality (LSI). The LSI constant is uniform in the boundary data, the external field and scales optimally in the system size. This extends a classical result of H.T. Yau from discrete to unbounded, real-valued spins. It also extends prior results of Landim et al. or Menz for unbounded, real-valued spins from absent- or weak- to strong-interaction. We deduce the LSI by combining two competing methods, the two-scale approach and the Zegarlinski method. Main ingredients are the strict convexity of the coarse-grained Hamiltonian, the equivalence of ensembles and the decay of correlations in the ce.

Logarithmic Sobolev inequalities in discrete product spaces

AbstractThe aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: for a fixed probability measure q on , ( is a finite set), and any probability measure on , (*) $$D(p||q){\rm{\le}}C \cdot \sum\limits_{i = 1}^n {{\rm{\mathbb{E}}}_p D(p_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,...,{\rm{ }}Y_n )||q_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,{\rm{ }}...,{\rm{ }}Y_n )),} $$ where pi(· |y1, …, yi−1, yi+1, …, yn) and qi(· |x1, …, xi−1, xi+1, …, xn) denote the local specifications for p resp. q, that is, the conditional distributions of the ith coordinate, given the other coordinates. The constant C depends on (the local specifications of) q.The inequality (*) ismeaningful in product spaces, in both the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for q, provided uniform logarithmic Sobolev inequalities are available for qi(· |x1, …, xi−1, xi+1, …, xn), for all fixed i and fixed (x1, …, xi−1, xi+1, …, xn). Inequality (*) directly implies that the Gibbs sampler associated with q is a contraction for relative entropy.In this paper we derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance.

A very simple proof of the LSI for high temperature spin systems

We present a very simple proof that the O(n) model satisfies a uniform logarithmic Sobolev inequality (LSI) if the difference between the largest and the smallest eigenvalue of the coupling matrix is less than n. This condition applies in particular to the SK spin glass model at inverse temperature β<1/4. It is the first result of rapid relaxation for the SK model and requires significant cancellations between the ferromagnetic and anti-ferromagnetic spin couplings that cannot be obtained by existing methods to prove Log-Sobolev inequalities. The proof also applies to more general bounded and unbounded spin systems. It uses a single step of zero range renormalisation and Bakry–Emery theory for the renormalised measure.

The Logarithmic Sobolev and Sobolev Inequalities Along the Ricci Flow

Based on Perelman’s entropy monotonicity, uniform logarithmic Sobolev inequalities along the Ricci flow are derived. Then uniform Sobolev inequalities along the Ricci flow are derived via harmonic analysis of the integral transform of the relevant heat operator. These inequalities are fundamental analytic properties of the Ricci flow. They are also extended to the volume-normalized Ricci flow and the Kahler–Ricci flow.

The Logarithmic Sobolev Inequality Along the Ricci Flow: The Case $$\lambda _0(g_0)=0$$ λ 0 ( g 0 ) = 0

A uniform logarithmic Sobolev inequality, a uniform Sobolev inequality and a uniform \(\kappa \)-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial metric is zero.

Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential

We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Villani, Westdickenberg and the second author from the quadratic to the general case. Using an asymmetric Brascamp-Lieb-type inequality for covariances, we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained Hamiltonian, which follows from a general local Cram\'{e}r theorem.

Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures

Une inegalite de Brascamp et Lieb donne une estimation sur la covariance entre deux fonctions par rapport a une mesure log-concave, qui est bornee par le produit des normes $L^{2}$ des gradients des fonctions, ou l’amplitude du gradient est calculee en utilisant un produit scalaire egal a l’inverse de la matrice Hessienne du potentiel de la mesure log-concave. Menz et Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. (2011) Preprint] ont prouve une variante de ce resultat ou les normes $L^{2}$ sont remplacees par des normes $L^{1}$ et $L^{\infty}$, mais seulement dans $\mathbb{R}^{1}$. Nous prouvons une generalisation de ces deux resultats, avec une extension de ces inegalites a des normes $L^{p}$ et $L^{q}$ dans $\mathbb{R}^{n}$, pour tout $n\geq1$. Nous prouvons aussi une inegalite pour des integrales de differences divisees de fonctions a l’aide des integrales de leurs gradients.

Uniform Logarithmic Sobolev Inequality for Boltzmann Measures with Exterior Magnetic Field over Spheres

In this paper, we obtain the uniform logarithmic Sobolev inequality for the Boltzmann measures by reducing multi-dimensional measures to one-dimensional measures, and then applying the characterization on the constant of logarithmic Sobolev inequality for a probability measure on the real line.

Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields

We show that the entropy functional exhibits a quasi-factorization property with respect to a pair of weakly dependent σ-algebras. As an application we give a simple proof that the Dobrushin and Shlosmans complete analyticity condition, for a Gibbs specification with finite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several different 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.

Remarks on decay of correlations and Witten Laplacians III. Application to logarithmic Sobolev inequalities

This is the continuation of our two previous articles devoted to the use of Witten Laplacians for analyzing Laplace integrals in statistical mechanics. The main application treated in Part I was a semi-classical one. The second application was more perturbative in spirit and gave very explicit estimates for the lower bound of the Witten Laplacian in the case of a quartic model. We shall relate in this third part our studies of the Witten Laplacian with the existence of uniform logarithmic Sobolev inequalities through a criterion of B. Zegarlinski. More precisely, our main contribution is to show how to control the decay of correlations uniformly with respect to various parameters, under a natural condition of strict convexity at ∞ of the single-spin phase and when the nearest neighbor interaction is small enough.

The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition

Given a finite range lattice gas with a compact, continuous spin space, it is shown (cf. Theorem 1.2) that a uniform logarithmic Sobolev inequality (cf. 1.4) holds if and only if the Dobrushin-Shlosman mixing condition (cf. 1.5) holds. As a consequence of our considerations, we also show (cf. Theorems 3.2 and 3.6) that these conditions are equivalent to a statement about the uniform rate at which the associated Glauber dynamics tends to equilibrium. In this same direction, we show (cf. Theorem 3.19) that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the Glauber dynamics.