Finite - size and Geometrical E ects in the Random

We study the transport properties of the percolating cluster for two diierent geometries. For the usual parallel bar geometry, we study nite-size eeects and report evidence that in the innnite size limit the entire backbone contributes to the low current. For the geometry in which the voltage diierence is applied to two given points separated by a distance r (in a system of size L), we propose a scaling form for the moments of the current and nd excellent agreement with simulations. This scaling form implies that at xed system size, the current is multifractal, but the (innnite) set of exponents is diierent from the usual case of parallel bars. 1 The transport properties of the percolating cluster have been the subject of numerous studies 1] for roughly twenty years. A particularly interesting system is the random resistor network (RRN), where the bonds have a random conductance. It was realised very early that the random resistor network could serve as a paradigm for transport properties in heterogeneous systems ]. The rst studies were devoted to eeective properties of the network (conductivity, per-mittivity, etc.) 2,3], but for many practical applications (fracture, dielectric breakdown 4]), the central quantity in this problem is the current probability distribution P(i). For instance , in the random fuse network, it is the maximum current (the hottest or \red" bonds) which will determine the macroscopic failure of the system ]. The probability distribution P(i) has many interesting features, one of which is multi-fractality 5{7]: in order to describe P(i), an innnite set of exponents is needed. In contrast to what happens on hierarchical lattices 6], the low current part of P(i) (as well as the mul-tifractal spectrum) has not a log-normal but rather a power-law behavior. There are very strong nite-size eeects and the exponent of the low-current part has a 1= ln L dependence 10], and the question of the asymptotic regime remains open so far. These results were obtained using only one geometry, namely parallel bars. In this case one looks for the percolation cluster (and the backbone) which connects the two sides of the system. However, another interesting situation was suggested in 8]. In this case, one looks at the backbone connecting two points separated by a distance r. In the following, we will refer to this situation as the \two injection points" geometry. This situation has a direct practical application in transport. For …