Electronic Journal of Probability

The approach of Otto-Reznikoff revisited

Georg Menz

Full-text: Open access


In this article we consider a lattice system of unbounded continuous spins. Otto and Reznikoff used the two-scale approach to show that exponential decay of correlations yields a logarithmic Sobolev inequality (LSI) with uniform constant in the system size. We improve their statement by weakening the assumptions, for which a more detailed analysis based on two new ingredients is needed. The two new ingredients are a covariance estimate and a uniform moment estimate. We additionally provide a comparison principle for covariances showing that the correlations of a conditioned Gibbs measure are controlled by the correlations of the original Gibbs measure with ferromagnetic interaction. This comparison principle simplifies the verification of the hypotheses of the main result. As an application of the main result we show how sufficient algebraic decay of correlations yields the uniqueness of the infinite-volume Gibbs measure, generalizing a result of Yoshida from finite-range to infinite-range interaction.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 107, 27 pp.

Accepted: 6 November 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82C26: Dynamic and nonequilibrium phase transitions (general)

lattice systems continuous spin logarithmic Sobolev inequality decay of correlations

This work is licensed under a Creative Commons Attribution 3.0 License.


Menz, Georg. The approach of Otto-Reznikoff revisited. Electron. J. Probab. 19 (2014), paper no. 107, 27 pp. doi:10.1214/EJP.v19-3418. https://projecteuclid.org/euclid.ejp/1465065749

Export citation


  • Bakry, D.; Émery, Michel. Diffusions hypercontractives. (French) [Hypercontractive diffusions] Séminaire de probabilités, XIX, 1983/84, 177–206, Lecture Notes in Math., 1123, Springer, Berlin, 1985.
  • Béllissard, Jean; Høegh-Krohn, Raphael. Compactness and the maximal Gibbs state for random Gibbs fields on a lattice. Comm. Math. Phys. 84 (1982), no. 3, 297–327.
  • Bodineau, T.; Helffer, B. The log-Sobolev inequality for unbounded spin systems. J. Funct. Anal. 166 (1999), no. 1, 168–178.
  • Dizdar, D. Schritte zu einer optimalen Konvergenzrate im hydrodynamischen Limes der Kawasaki Dynamik (towards an optimal rate of convergence in the hydrodynamic limit for Kawasaki dynamics), Ph.D. thesis, Diploma thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, 2007.
  • Gross, Leonard. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061–1083.
  • Grunewald, Natalie; Otto, Felix; Villani, Cédric; Westdickenberg, Maria G. A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 2, 302–351.
  • Holley, Richard; Stroock, Daniel. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46 (1987), no. 5-6, 1159–1194.
  • Horiguchi, T.; Morita, T. Upper and lower bounds to a correlation function for an Ising model with random interactions. Phys. Lett. A 74 (1979), no. 5, 340–342.
  • Ledoux, M. Logarithmic Sobolev inequalities for unbounded spin systems revisited. Séminaire de Probabilités, XXXV, 167–194, Lecture Notes in Math., 1755, Springer, Berlin, 2001.
  • Martinelli, F.; Olivieri, E. Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Comm. Math. Phys. 161 (1994), no. 3, 447–486.
  • Menz, Georg; A Brascamp-Lieb type covariance estimate, Electron. J. Probab. 19 (2014), no. 78, 1–15.
  • Menz, Georg; Nittka, Robin. Decay of correlations in 1D lattice systems of continuous spins and long-range interaction. J. Stat. Phys. 156 (2014), no. 2, 239–267.
  • Menz, Georg; Otto, Felix. Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Ann. Probab. 41 (2013), no. 3B, 2182–2224.
  • Otto, Felix; Reznikoff, Maria G. A new criterion for the logarithmic Sobolev inequality and two applications. J. Funct. Anal. 243 (2007), no. 1, 121–157.
  • Procacci, Aldo; Scoppola, Benedetto. On decay of correlations for unbounded spin systems with arbitrary boundary conditions. J. Statist. Phys. 105 (2001), no. 3-4, 453–482.
  • Royer, Gilles. An initiation to logarithmic Sobolev inequalities. Translated from the 1999 French original by Donald Babbitt. SMF/AMS Texts and Monographs, 14. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2007. viii+119 pp. ISBN: 978-0-8218-4401-4; 0-8218-4401-6
  • Stroock, Daniel W.; Zegarliński, Bogusław. The logarithmic Sobolev inequality for continuous spin systems on a lattice. J. Funct. Anal. 104 (1992), no. 2, 299–326.
  • Stroock, Daniel W.; Zegarliński, Bogusław. The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 (1992), no. 1, 175–193.
  • Sylvester, Garrett S. Inequalities for continuous-spin Ising ferromagnets. J. Statist. Phys. 15 (1976), no. 4, 327–341.
  • Yoshida, Nobuo. The log-Sobolev inequality for weakly coupled lattice fields. Probab. Theory Related Fields 115 (1999), no. 1, 1–40.
  • Yoshida, Nobuo. The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 2, 223–243.
  • Yoshida, Nobuo. Phase transition from the viewpoint of relaxation phenomena. Rev. Math. Phys. 15 (2003), no. 7, 765–788.
  • Zegarlinski, Boguslaw. The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Comm. Math. Phys. 175 (1996), no. 2, 401–432.
  • Zitt, Pierre-André. Functional inequalities and uniqueness of the Gibbs measure - from log-Sobolev to Poincaré. ESAIM Probab. Stat. 12 (2008), 258–272.