Illinois Journal of Mathematics

Long range correlation inequalities for massless Euclidean fields

Joseph G. Conlon and Arash Fahim

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, new correlation inequalities are obtained for massless Euclidean fields on the $d$ dimensional integer lattice. Some of the inequalities have been obtained previously, in the case where the Lagrangian is a very small perturbation of a quadratic, using the renormalization group method. The results of the present paper apply provided the Lagrangian is uniformly convex. They therefore hold for the Coulomb dipole gas in which particle density can be of order $1$. The approach of the present paper is based on the methodology of Naddaf–Spencer, which relates second moment correlation functions for the Euclidean field to expectations of Green’s functions for parabolic PDE with random coefficients.

Article information

Illinois J. Math., Volume 59, Number 1 (2015), 143-187.

Received: 7 April 2015
Revised: 26 August 2015
First available in Project Euclid: 11 February 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B28: Renormalization group methods [See also 81T17]


Conlon, Joseph G.; Fahim, Arash. Long range correlation inequalities for massless Euclidean fields. Illinois J. Math. 59 (2015), no. 1, 143--187. doi:10.1215/ijm/1455203163.

Export citation


  • D. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. (N.S.) 73 (1967), 890–896.
  • G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization, Asymptot. Anal. 59 (2008), 1–26.
  • C. Boldrighini, R. Minlos and A. Pellegrinotti, Random walks in quenched i.i.d. space–time environments are always a.s. diffusive, Probab. Theory Related Fields 129 (2004), 133–156.
  • H. Brascamp and E. Lieb, On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), 366–389.
  • D. Brydges, Lectures on the renormalisation group, Statistical mechanics, IAS/Park City Math. Ser., vol. 16, Amer. Math. Soc., Providence, 2009, pp. 7–93.
  • D. Brydges and H.-T. Yau, Grad $\phi$ perturbations of massless Gaussian fields, Comm. Math. Phys. 129 (1990), 351–392.
  • L. Caffarelli and P. Souganidis, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math. 180 (2010), 301–360.
  • R. Carmona and M. Tehranchi, Interest rate models: An infinite dimensional stochastic analysis perspective, Springer-Verlag, Berlin–Heidelberg, 2006.
  • G. Checkin, A. Piatnitski and A. Shamaev, Homogenization: Methods and applications, Translations of Mathematical Monographs, vol. 234, Amer. Math. Soc., Providence, 2007.
  • J. Conlon, PDE with random coefficients and Euclidean field theories, J. Stat. Phys. 114 (2004), 933–958.
  • J. Conlon, Greens functions for elliptic and parabolic equations with random coefficients II, Trans. Amer. Math. Soc. 356 (2004), 4085–4142.
  • J. Conlon and A. Fahim, Strong convergence to the homogenized limit of parabolic equations with random coefficients, Trans. Amer. Math. Soc. 367 (2015), 3041–3093.
  • J. Conlon and A. Fahim, Strong convergence to the homogenized limit of elliptic equations with random coefficients II, Bull. Lond. Math. Soc. 45 (2013), 973–986.
  • J. Conlon and T. Spencer, Strong convergence to the homogenized limit of elliptic equations with random coefficients, Trans. Amer. Math. Soc. 366 (2014), 1257–1288.
  • T. Delmotte and J. Deuschel, On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nabla\phi$ interface model, Probab. Theory Related Fields 133 (2005), 358–390.
  • J. Dimock, Infinite volume limit for the dipole gas, J. Stat. Phys. 135 (2009), 393–427.
  • J. Dimock and T. Hurd, A renormalization group analysis of correlation functions for the dipole gas, J. Stat. Phys. 66 (1992), 1277–1318.
  • D. Dolgopyat, G. Keller and C. Liverani, Random walk in Markovian environment, Ann. Probab. 36 (2008), 1676–1710.
  • J. Fröhlich and Y. M. Park, Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems, Comm. Math. Phys. 59 (1978), 235–266.
  • T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg–Landau $\nabla \phi$ interface model, Comm. Math. Phys. 185 (1997), 1–36.
  • K. Gawedzki and A. Kupiainen, Lattice dipole gas and $(\nabla\phi)^4$ models at long distances: Decay of correlations and scaling limit, Comm. Math. Phys. 92 (1984), 531–553.
  • G. Giacomin, S. Olla and H. Spohn, Equilibrium fluctuations for $\nabla\phi$ interface model, Ann. Probab. 29 (2001), 1138–1172.
  • A. Gloria and F. Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab. 39 (2011), 779–856.
  • A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab. 22 (2012), 1–28.
  • A. Gloria, S. Neukamm and F. Otto, Quantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on glauber dynamics, Invent. Math. 199 (2015), 455–515.
  • B. F. Jones, A class of singular integrals, Amer. J. Math. 86 (1964), 441–462.
  • I. Karatzas and S. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991.
  • S. Kozlov, Averaging of random structures, Dokl. Akad. Nauk SSSR 241 (1978), 1016–1019.
  • S. Kozlov, The method of averaging and walks in inhomogeneous environment, Russian Math. Surveys 40 (1985), 73–145.
  • C. Landim, S. Olla and H. T. Yau, Convection–diffusion equation with space–time ergodic random flow, Probab. Theory Related Fields 112 (1998), 203–220.
  • D. Marahrens and F. Otto, Annealed estimates on the Green function, Probab. Theory Related Fields 163 (2015), 527–573.
  • N. Meyers, An $L^p$ estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (1963), 189–206.
  • J. Moser, On Harnack's inequality for parabolic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591.
  • J. C. Mourrat, Kantorivich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients, Probab. Theory Related Fields 160 (2014), 279–314.
  • D. Mugler, Green's functions for the finite difference heat, Laplace and wave equations, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983), Internat. Schriftenreihe Numer. Math., vol. 65, Birkhäuser, Basel, 1984, pp. 543–554.
  • A. Naddaf and T. Spencer, On homogenization and scaling limit of some gradient perturbations of a massless free field, Comm. Math. Phys. 183 (1997), 55–84.
  • A. Naddaf and T. Spencer, Estimates on the variance of some homogenization problems, preprint, 1998.
  • D. Nualart, The Malliavin calculus and related topics, 2nd ed., Springer Verlag, Berlin, 2005.
  • G. Papanicolaou and S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, vol. 2, Coll. Math. Soc. Janos Bolya, vol. 27, North Holland, Amsterdam, 1981, pp. 835–873.
  • R. Rhodes, On homogenization of space-time dependent and degenerate random flows, Stochastic Process. Appl. 117 (2007), 1561–1585.
  • E. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, NJ, 1970.
  • V. Yurinskii, Averaging of symmetric diffusion in random medium, Sibirsk. Mat. Zh. 27 (1986), 167–180.
  • V. Zhikov, S. Kozlov and O. Oleinik, Homogenization of differential operators and integral functionals, Springer Verlag, Berlin, 1994.