The Annals of Probability

Noise-stability and central limit theorems for effective resistance of random electric networks

Raphaël Rossignol

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Abstract

We investigate the (generalized) Walsh decomposition of point-to-point effective resistances on countable random electric networks with i.i.d. resistances. We show that it is concentrated on low levels, and thus point-to-point effective resistances are uniformly stable to noise. For graphs that satisfy some homogeneity property, we show in addition that it is concentrated on sets of small diameter. As a consequence, we compute the right order of the variance and prove a central limit theorem for the effective resistance through the discrete torus of side length $n$ in $\mathbb{Z}^{d}$, when $n$ goes to infinity.

Article information

Source
Ann. Probab., Volume 44, Number 2 (2016), 1053-1106.

Dates
Received: June 2012
Revised: June 2014
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1457960391

Digital Object Identifier
doi:10.1214/14-AOP996

Mathematical Reviews number (MathSciNet)
MR3474467

Zentralblatt MATH identifier
1347.60133

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C21: Flows in graphs

Keywords
Effective resistance conductance noise sensitivity and stability Efron–Stein inequality generalized Walsh decomposition central limit theorem stochastic homogenization

Citation

Rossignol, Raphaël. Noise-stability and central limit theorems for effective resistance of random electric networks. Ann. Probab. 44 (2016), no. 2, 1053--1106. doi:10.1214/14-AOP996. https://projecteuclid.org/euclid.aop/1457960391


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References

  • [1] Benaïm, M. and Rossignol, R. (2008). Exponential concentration for first passage percolation through modified Poincaré inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 44 544–573.
  • [2] Benjamini, I., Kalai, G. and Schramm, O. (1999). Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90 5–43.
  • [3] Benjamini, I. and Rossignol, R. (2008). Submean variance bound for effective resistance of random electric networks. Comm. Math. Phys. 280 445–462.
  • [4] Biskup, M., Salvi, M. and Wolff, T. (2014). A central limit theorem for the effective conductance: Linear boundary data and small ellipticity contrasts. Comm. Math. Phys. 328 701–731.
  • [5] Boivin, D. (2009). Tail estimates for homogenization theorems in random media. ESAIM Probab. Stat. 13 51–69.
  • [6] Boivin, D. and Depauw, J. (2003). Spectral homogenization of reversible random walks on $\mathbb{Z}^{d}$ in a random environment. Stochastic Process. Appl. 104 29–56.
  • [7] Bollobás, B. (1998). Modern Graph Theory. Graduate Texts in Mathematics 184. Springer, New York.
  • [8] Bourgain, J. (1980). Walsh subspaces of $L^{p}$-product spaces. In Seminar on Functional Analysis, 19791980 (French) Exp. No. 4A, 9. École Polytech., Palaiseau.
  • [9] Caputo, P. and Ioffe, D. (2003). Finite volume approximation of the effective diffusion matrix: The case of independent bond disorder. Ann. Inst. Henri Poincaré Probab. Stat. 39 505–525.
  • [10] Chatterjee, S. (2008). Chaos, concentration, and multiple valleys. Available at arXiv:0810.4221.
  • [11] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
  • [12] Chen, L. H. Y. and Shao, Q.-M. (2004). Normal approximation under local dependence. Ann. Probab. 32 1985–2028.
  • [13] Delmotte, T. (1997). Inégalité de Harnack elliptique sur les graphes. Colloq. Math. 72 19–37.
  • [14] Delmotte, T. and Deuschel, J.-D. (2005). On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nabla\phi$ interface model. Probab. Theory Related Fields 133 358–390.
  • [15] Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586–596.
  • [16] Gloria, A. and Otto, F. (2011). An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 779–856.
  • [17] Gloria, A. and Otto, F. (2012). An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 1–28.
  • [18] Hatami, H. (2012). A structure theorem for Boolean functions with small total influences. Ann. of Math. (2) 176 509–533.
  • [19] Jikov, V. V., Kozlov, S. M. and Oleĭnik, O. A. (1994). Homogenization of Differential Operators and Integral Functionals. Springer, Berlin.
  • [20] Kirchhoff, G. (1958). On the solution of the equations obtained from the investigation of the linear distribution of galvanic currents. IRE Trans. Circuit Theory 5, 4–7.
  • [21] Kozlov, S. M. (1986). Average difference schemes. Mat. Sb. (N.S.) 129, 171 338–357, 447.
  • [22] Künnemann, R. (1983). The diffusion limit for reversible jump processes on $\mathbf{Z}^{d}$ with ergodic random bond conductivities. Comm. Math. Phys. 90 27–68.
  • [23] Lyons, R. and Peres, Y. (2014). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge.
  • [24] Moser, J. (1961). On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 577–591.
  • [25] Mossel, E. (2010). Gaussian bounds for noise correlation of functions. Geom. Funct. Anal. 19 1713–1756.
  • [26] Naddaf, A. and Spencer, T. (1998). Estimates on the variance of some homogenization problems. Unpublished.
  • [27] Nolen, J. (2014). Normal approximation for a random elliptic equation. Probab. Theory Related Fields 159 661–700.
  • [28] Owhadi, H. (2003). Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Related Fields 125 225–258.
  • [29] Papanicolaou, G. C. and Varadhan, S. R. S. (1981). Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979). Colloquia Mathematica Societatis János Bolyai 27 835–873. North-Holland, Amsterdam.
  • [30] Rosenthal, H. P. (1970). On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables. Israel J. Math. 8 273–303.
  • [31] Wehr, J. (1997). A lower bound on the variance of conductance in random resistor networks. J. Stat. Phys. 86 1359–1365.