Breakdown properties of quenched random systems: The random-fuse network

An analysis of a prototypical percolation model (the fuse network) for breakdown in quenched random systems is given. The breakdown voltage and the topology of the eventual breakdown path are studied analytically and numerically. New scaling concepts, based on the most critical defect in the network, combined with standard percolation scaling ideas, lead to a complete picture of the strength of the network. The mean breakdown strength and the distribution of breakdown strengths are derived in the different concentration regimes. The breakdown path is described by new order parameters on approach to $p_c$. One, the number of bonds broken in the breakdown process, is studied in detail. Many models and physical systems should show an analogous behavior and simplified models for two of these problems, brittle fracture and dielectric breakdown in solids, are discussed.