Tail estimates for homogenization theorems in random media

Daniel Boivin

ESAIM: Probability and Statistics (2009)

  • Volume: 13, page 51-69
  • ISSN: 1292-8100

Abstract

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Consider a random environment in d given by i.i.d. conductances. In this work, we obtain tail estimates for the fluctuations about the mean for the following characteristics of the environment: the effective conductance between opposite faces of a cube, the diffusion matrices of periodized environments and the spectral gap of the random walk in a finite cube.

How to cite

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Boivin, Daniel. "Tail estimates for homogenization theorems in random media." ESAIM: Probability and Statistics 13 (2009): 51-69. <http://eudml.org/doc/250668>.

@article{Boivin2009,
abstract = { Consider a random environment in $\{\mathbb Z\}^d$ given by i.i.d. conductances. In this work, we obtain tail estimates for the fluctuations about the mean for the following characteristics of the environment: the effective conductance between opposite faces of a cube, the diffusion matrices of periodized environments and the spectral gap of the random walk in a finite cube. },
author = {Boivin, Daniel},
journal = {ESAIM: Probability and Statistics},
keywords = {Periodic approximation; random environments; fluctuations; effective diffusion matrix; effective conductance; non-uniform ellipticity; non-uniform ellipticity},
language = {eng},
month = {2},
pages = {51-69},
publisher = {EDP Sciences},
title = {Tail estimates for homogenization theorems in random media},
url = {http://eudml.org/doc/250668},
volume = {13},
year = {2009},
}

TY - JOUR
AU - Boivin, Daniel
TI - Tail estimates for homogenization theorems in random media
JO - ESAIM: Probability and Statistics
DA - 2009/2//
PB - EDP Sciences
VL - 13
SP - 51
EP - 69
AB - Consider a random environment in ${\mathbb Z}^d$ given by i.i.d. conductances. In this work, we obtain tail estimates for the fluctuations about the mean for the following characteristics of the environment: the effective conductance between opposite faces of a cube, the diffusion matrices of periodized environments and the spectral gap of the random walk in a finite cube.
LA - eng
KW - Periodic approximation; random environments; fluctuations; effective diffusion matrix; effective conductance; non-uniform ellipticity; non-uniform ellipticity
UR - http://eudml.org/doc/250668
ER -

References

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  1. I. Benjamini and R. Rossignol, Submean variance bound for effective resistance on random electric networks.v4 [math.PR]  Zbl1207.82024URIarXiv:math/0610393
  2. D. Boivin and J. Depauw, Spectral homogenization of reversible random walks on d in a random environment. Stochastic Process. Appl.104 (2003) 29–56.  Zbl1075.35507
  3. D. Boivin and Y. Derriennic, The ergodic theorem for additive cocycles of d or d . Ergodic Theory Dynam. Syst.11 (1991) 19–39.  Zbl0723.60008
  4. E. Bolthausen and A.S. Sznitman, Ten lectures on random media. DMV Seminar, Band 32, Birkhäuser (2002).  Zbl1075.60128
  5. A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré Probab. Statist.40 (2004) 153–165.  Zbl1058.35023
  6. P. Caputo and D. Ioffe, Finite volume approximation of the effective diffusion matrix: the case of independent bond disorder. Ann. Inst. H. Poincaré Probab. Statist.39 (2003) 505–525.  Zbl1014.60094
  7. F.R.K. Chung, Spectral graph theory. CBMS Regional Conference Series in Mathematics, 92. American Mathematical Society (1997).  Zbl0867.05046
  8. E.B. Davies, Heat kernels and spectral theory. Cambridge Tracts in Mathematics, 92. Cambridge University Press (1989).  Zbl0699.35006
  9. T. Delmotte, Inéalité de Harnack elliptique sur les graphes. Colloq. Math.72 (1997) 19–37.  Zbl0871.31008
  10. R. Durrett, Probability: Theory and Examples. Wadsworth & Brooks/Cole Statistics/Probability Series (1991).  Zbl0709.60002
  11. L.R.G. Fontes and P. Mathieu, On symmetric random walks with random conductances on d . Probab. Theory Related Fields134 (2006) 565–602.  Zbl1086.60066
  12. T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau φ interface model. Commun. Math. Phys. 185 (1997) 1–36.  Zbl0884.58098
  13. G. Grimmett, Percolation. 2nd ed. Springer (1999).  
  14. E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes Mathematics 5. American Mathematical Society (2000).  Zbl0981.58006
  15. B.D. Hughes, Random walks and random environments. Vol. 2. Random environments. Oxford University Press (1996).  Zbl0925.60076
  16. V.V. Jikov, S.M. Kozlov and O.A. Olejnik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994).  
  17. S. Kesavan, Homogenization of elliptic eigenvalue problems I. Appl. Math. Optimization5 (1979) 153–167.  Zbl0415.35061
  18. H. Kesten, On the speed of convergence in first-passage percolation. Ann. Appl. Probab.3 (1993) 296–338.  Zbl0783.60103
  19. C. Kipnis and S.R.S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys.10 (1986) 1–19.  Zbl0588.60058
  20. S.M. Kozlov, The method of averaging and walks in inhomogeneous environments. Russ. Math. Surv.40 (1985) 73–145.  Zbl0615.60063
  21. R. Kuennemann, The diffusion limit for reversible jump processes on Z m with ergodic random bond conductivities. Commun. Math. Phys.90 (1983) 27–68.  Zbl0523.60097
  22. H. Owhadi, Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Related Fields125 (2003) 225-258.  Zbl1040.60025
  23. E. Pardoux and A.Yu. Veretennikov, On the Poisson equation and diffusion approximation. I. Ann. Probab.29 (2001) 1061–1085.  Zbl1029.60053
  24. Y. Peres, Probability on trees: An introductory climb. Lectures on probability theory and statistics. École d'été de Probabilités de Saint-Flour XXVII-1997, Springer. Lect. Notes Math. 1717 (1999) 193–280 .  
  25. L. Saloff-Coste, Lectures on finite Markov chains. Lectures on probability theory and statistics. École d'été de probabilités de Saint-Flour XXVI–1996, Springer. Lect. Notes Math. 1665 (1997) 301–413.  
  26. V. Sidoravicius and A.-S. Sznitman, Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields129 (2004) 219–244.  Zbl1070.60090
  27. F. Spitzer, Principles of random walk. The University Series in Higher Mathematics. D. Van Nostrand Company (1964).  Zbl0119.34304
  28. J. Wehr, A lower bound on the variance of conductance in random resistor networks. J. Statist. Phys.86 (1997) 1359–1365.  Zbl0937.82024
  29. V.V. Yurinsky, Averaging of symmetric diffusion in random medium. Sib. Math. J.2 (1986) 603–613.  Zbl0614.60051

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