## The Annals of Probability

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

Raphaël Rossignol

#### 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
Revised: June 2014
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.aop/1457960391

Digital Object Identifier
doi:10.1214/14-AOP996

Mathematical Reviews number (MathSciNet)
MR3474467

Zentralblatt MATH identifier
1347.60133

#### 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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