Journal of Statistical Physics, Vol. 69, Nos. 5/6, 1992 Sample-to-Sample Fluctuations in the Conductivity of a Disordered Medium Till Schl6sser 1 and Herbert Spohn 1 Received December 31, 1991," final April 24, 1992 We investigate the sample-to-sample fluctuations in the conductivity of a ran- dom resistor network--equivalently, in the diffusivity of a disordered medium with symmetric hopping rates. We argue that whenever the effective conduc- tivity ~* 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 a* vanishes when approaching the percolation threshold Pc. Close to Pc the fluctua- tions are anomalous. From the renormalization group on hierarchical lattices we find that at Pc fluctuations and mean scale in the same fashion, i.e., there is no independent scaling exponent for the fluctuations. KEY WORDS: Random resistor network; percolation threshold; scaling exponents. 1. INTRODUCTION It has been recognized for some time that sample-to-sample fluctuations are an important issue for disordered systems. If the quantity of physical interest is not self-averaging as the size of the system becomes large, then a mere disorder average may be meaningless and the full distribution must be elucidated. But even for quantities which are self-averaging the fluctua- tions (with respect to disorder) carry information on qualitative changes in the behavior of the 'system. For example, the free energy of the Sherrington-Kirkpatrick model of a spin glass is self-averaging, but fluctuations change when passing through T,.. (1) In this paper we investigate the sample-to-sample fluctuations in the conductivity of a random resistor network (equivalently in the diffusivity t Theoretische Physik, Universit/it M/inchen, D-8000 Miinchen 2, Germany. 955 0022-4715/92/1200-0955506.50/0 9 1992 Plenum Publishing Corporation