Tail estimates for homogenization theorems in random media
ESAIM: Probability and Statistics, Volume 13  (2009), p. 51-69

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.

DOI : https://doi.org/10.1051/ps:2007036
Classification:  60K37,  35B27,  82B44
Keywords: periodic approximation, random environments, fluctuations, effective diffusion matrix, effective conductance, non-uniform ellipticity
@article{PS_2009__13__51_0,
     author = {Boivin, Daniel},
     title = {Tail estimates for homogenization theorems in random media},
     journal = {ESAIM: Probability and Statistics},
     publisher = {EDP-Sciences},
     volume = {13},
     year = {2009},
     pages = {51-69},
     doi = {10.1051/ps:2007036},
     zbl = {pre05660758},
     mrnumber = {2493855},
     language = {en},
     url = {http://www.numdam.org/item/PS_2009__13__51_0}
}
Boivin, Daniel. Tail estimates for homogenization theorems in random media. ESAIM: Probability and Statistics, Volume 13 (2009) , pp. 51-69. doi : 10.1051/ps:2007036. http://www.numdam.org/item/PS_2009__13__51_0/

[1] I. Benjamini and R. Rossignol, Submean variance bound for effective resistance on random electric networks. arXiv:math/0610393v4 [math.PR] | MR 2395478 | Zbl pre05306109

[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. | MR 1956471 | Zbl 1075.35507

[3] D. Boivin and Y. Derriennic, The ergodic theorem for additive cocycles of d or d . Ergodic Theory Dynam. Syst. 11 (1991) 19-39. | MR 1101082 | Zbl 0723.60008

[4] E. Bolthausen and A.S. Sznitman, Ten lectures on random media. DMV Seminar, Band 32, Birkhäuser (2002). | MR 1890289 | Zbl 1075.60128

[5] A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 153-165. | Numdam | MR 2044813 | Zbl 1058.35023

[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. | Numdam | MR 1978989 | Zbl 1014.60094

[7] F.R.K. Chung, Spectral graph theory. CBMS Regional Conference Series in Mathematics, 92. American Mathematical Society (1997). | MR 1421568 | Zbl 0867.05046

[8] E.B. Davies, Heat kernels and spectral theory. Cambridge Tracts in Mathematics, 92. Cambridge University Press (1989). | MR 990239 | Zbl 0699.35006

[9] T. Delmotte, Inéalité de Harnack elliptique sur les graphes. Colloq. Math. 72 (1997) 19-37. | MR 1425544 | Zbl 0871.31008

[10] R. Durrett, Probability: Theory and Examples. Wadsworth & Brooks/Cole Statistics/Probability Series (1991). | MR 1068527 | Zbl 0709.60002

[11] L.R.G. Fontes and P. Mathieu, On symmetric random walks with random conductances on d . Probab. Theory Related Fields 134 (2006) 565-602. | MR 2214905 | Zbl 1086.60066

[12] T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau φ interface model. Commun. Math. Phys. 185 (1997) 1-36. | MR 1463032 | Zbl 0884.58098

[13] G. Grimmett, Percolation. 2nd ed. Springer (1999). | MR 1707339 | Zbl 0926.60004

[14] E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes Mathematics 5. American Mathematical Society (2000). | MR 1688256 | Zbl 0981.58006

[15] B.D. Hughes, Random walks and random environments. Vol. 2. Random environments. Oxford University Press (1996). | MR 1420619 | Zbl 0925.60076

[16] V.V. Jikov, S.M. Kozlov and O.A. Olejnik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994). | MR 1329546 | Zbl 0838.35001

[17] S. Kesavan, Homogenization of elliptic eigenvalue problems I. Appl. Math. Optimization 5 (1979) 153-167. | MR 533617 | Zbl 0415.35061

[18] H. Kesten, On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 (1993) 296-338. | MR 1221154 | Zbl 0783.60103

[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. | MR 834478 | Zbl 0588.60058

[20] S.M. Kozlov, The method of averaging and walks in inhomogeneous environments. Russ. Math. Surv. 40 (1985) 73-145. | MR 786087 | Zbl 0615.60063

[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. | MR 714611 | Zbl 0523.60097

[22] H. Owhadi, Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Related Fields 125 (2003) 225-258. | MR 1961343 | Zbl 1040.60025

[23] E. Pardoux and A.Yu. Veretennikov, On the Poisson equation and diffusion approximation. I. Ann. Probab. 29 (2001) 1061-1085. | MR 1872736 | Zbl 1029.60053

[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 . | MR 1746302 | Zbl 0957.60001

[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. | MR 1490046 | Zbl 0885.60061

[26] V. Sidoravicius and A.-S. Sznitman, Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2004) 219-244. | MR 2063376 | Zbl 1070.60090

[27] F. Spitzer, Principles of random walk. The University Series in Higher Mathematics. D. Van Nostrand Company (1964). | MR 171290 | Zbl 0119.34304

[28] J. Wehr, A lower bound on the variance of conductance in random resistor networks. J. Statist. Phys. 86 (1997) 1359-1365. | MR 1450770 | Zbl 0937.82024

[29] V.V. Yurinsky, Averaging of symmetric diffusion in random medium. Sib. Math. J. 2 (1986) 603-613. | Zbl 0614.60051