Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps

Franziska Flegel, Martin Heida, and Martin Slowik

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


We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in $\mathbb{Z}^{d}$. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum.

We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length.

Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for the normalized and rescaled local times of the random walk in a growing box.

Our proofs are based on a compactness result for the Laplacian’s Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence.


Nous considérons les propriétés d’homogénéisation de l’opérateur de Laplace discret avec des conductances aléatoires. Nous démontrons l’homogénéisation de l’équation de Poisson discrète et des plus hauts éléments du spectre de l’opérateur de Dirichlet dans un domaine limité.

Nous supposons les conductances stationnaires, ergodiques et strictement positives à plus proches voisins. Comparé aux résultats précédents, nous remplaçons l’ellipticité uniforme par des conditions d’intégrabilité des moments des conductances. De plus, nous autorisons des sauts de tailles arbitraires.

En l’absence de sauts longs, les conditions sur les moments sont optimales pour l’homogénéisation spectrale. Elles correspondent à la condition nécessaire du théorème central limite pour les marches aléatoires en conductances aléatoires. Nous utilisons l’homogénéisation spectrale pour démontrer un principe de grandes déviations gelé pour le temps local normalisé de la marche aléatoire dans une suite croissante de boites.

Nos démonstrations sont basées sur un résultat de compacité pour l’énergie de Dirichlet, les inégalités de Poincaré, l’itération de Moser et la convergence à deux échelles.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1226-1257.

Received: 9 February 2017
Revised: 23 March 2018
Accepted: 31 May 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60H25: Random operators and equations [See also 47B80] 60K37: Processes in random environments 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 47B80: Random operators [See also 47H40, 60H25] 47A75: Eigenvalue problems [See also 47J10, 49R05]

Random conductance model Homogenization Dirichlet eigenvalues Local times Percolation


Flegel, Franziska; Heida, Martin; Slowik, Martin. Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1226--1257. doi:10.1214/18-AIHP917.

Export citation


  • [1] S. Andres, J.-D. Deuschel and M. Slowik. Invariance principle for the random conductance model in a degenerate ergodic environment. Ann. Probab. 43 (4) (2015) 1866–1891.
  • [2] S. Andres, J.-D. Deuschel and M. Slowik. Harnack inequalities on weighted graphs and some applications to the random conductance model. Probab. Theory Related Fields 164 (3–4) (2016) 931–977.
  • [3] G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (6) (1992) 1482–1518.
  • [4] D. Boivin and J. Depauw. Spectral homogenization of reversible random walks on $\mathbb{Z}^{d}$ in a random environment. Stochastic Process. Appl. 104 (1) (2003) 29–56.
  • [5] M. Biskup, R. Fukushima and W. König. Eigenvalue fluctuations for lattice Anderson Hamiltonians. SIAM J. Math. Anal. 48 (4) (2016) 2674–2700.
  • [6] M. Biskup, R. Fukushima and W. König. Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials. Interdiscip. Inform. Sci. 25 (1) (2018) 59–76.
  • [7] J.-P. Bouchaud and A. Georges. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 195 (4–5) (1990) 127–293.
  • [8] M. Biskup. Recent progress on the random conductance model. Probab. Surv. 8 (2011) 294–373.
  • [9] H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York, 2011.
  • [10] T. Coulhon. Espaces de Lipschitz et inégalités de Poincaré. J. Funct. Anal. 136 (1) (1996) 81–113.
  • [11] J.-D. Deuschel, T. A. Nguyen and M. Slowik. Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights. Probab. Theory Related Fields 170 (1–2) (2018) 363–386.
  • [12] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, I. Comm. Pure Appl. Math. 28 (1975) 1–47.
  • [13] L. C. Evans. Partial Differential Equations, 2nd edition. Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI, 2010.
  • [14] A. Faggionato. Random walks and exclusion processes among random conductances on random infinite clusters: Homogenization and hydrodynamic limit. Electron. J. Probab. 13 (73) (2008) 2217–2247.
  • [15] A. Faggionato. Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models. Electron. J. Probab. 17 (15) (2012) 36pp.
  • [16] F. Flegel. Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model. Electron. J. Probab. 23 (68) (2018) 43pp.
  • [17] N. Gantert, W. König and Z. Shi. Annealed deviations of random walk in random scenery. Ann. Inst. Henri Poincaré Probab. Stat. 43 (1) (2007) 47–76.
  • [18] A. Giunti and J.-C. Mourrat. Quantitative homogenization of degenerate random environments. Ann. Inst. Henri Poincaré Probab. Stat. 54 (1) (2018) 22–50.
  • [19] V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994.
  • [20] H. Kesten. Aspects of first passage percolation. In École D’été de Probabilités de Saint-Flour, XIV – 1984 125–264. Lecture Notes in Math. 1180. Springer, Berlin, 1986.
  • [21] W. König and T. Wolff. Large deviations for the local times of a random walk among random conductances in a growing box. Markov Process. Related Fields 21 (3) (2015) 591–638.
  • [22] T.-W. Ma. Banach–Hilbert Spaces, Vector Measures and Group Representations. World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
  • [23] G. Dal Maso. An Introduction to $\Gamma$-Convergence. Progress in Nonlinear Differential Equations and Their Applications 8. Birkhäuser Boston, Inc., Boston, MA, 1993.
  • [24] P. Mathieu and A. Piatnitski. Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2085) (2007) 2287–2307.
  • [25] S. Neukamm, M. Schäffner and A. Schlömerkemper. Stochastic homogenization of nonconvex discrete energies with degenerate growth. SIAM J. Math. Anal. 49 (3) (2017) 1761–1809.
  • [26] G. Nguetseng. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (3) (1989) 608–623.
  • [27] A. Piatnitski and E. Zhizhina. Periodic homogenization of nonlocal operators with a convolution-type kernel. SIAM J. Math. Anal. 49 (1) (2017) 64–81.
  • [28] L. Saloff-Coste. Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics 301–413. Saint-Flour, 1996. Lecture Notes in Math. 1665. Springer, Berlin, 1997.
  • [29] E. Seneta. Non-negative Matrices and Markov Chains. Springer Series in Statistics. Springer, New York, 2006. Revised reprint of the second (1981) edition.
  • [30] V. V. Zhikov and A. L. Pyatnitskiĭ. Homogenization of random singular structures and random measures. Izv. Ross. Akad. Nauk Ser. Mat. 70 (1) (2006) 23–74.