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

TL;DR

This paper studies the stability and distributional properties of effective resistance in random electric networks, showing noise stability, variance estimates, and a central limit theorem for large toroidal grids.

Contribution

It introduces a Walsh decomposition approach to analyze effective resistance, demonstrating noise stability and deriving a central limit theorem for large-scale networks.

Findings

Effective resistance is stable under noise due to low-level Walsh decomposition.

Variance of effective resistance is of the right order for large networks.

A central limit theorem is proved for the effective resistance in high-dimensional tori.

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.