  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.
