Sample-to-sample fluctuations in the conductivity of a disordered medium

We investigate the sample-to-sample fluctuations in the conductivity of a random resistor network—equivalently, in the diffusivity of a disordered medium with symmetric hopping rates. We argue that whenever the effective conductivity ^* is strictly positive, then the fluctuations are normal, i.e., proportional to (volume)^–1/2. If the local conductivities are allowed to be zero, then ^* vanishes when approaching the percolation thresholdp _c. Close top _c the fluctuations are anomalous. From the renormalization group on hierarchical lattices we find that atp _c fluctuations and mean scale in the same fashion, i.e., there is no independent scaling exponent for the fluctuations.