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Two percolation models for electrical breakdown in quenched random media, a fuse-wire network and a dielectric network, are introduced and studied. A combination of Lifshitz and scaling arguments leads to a size dependence given by V b /L -a(p)/[l +M/?)(lnL )*], where 0= \/{d-1) for the fuse network and /3« 1 for the dielectric network. Simulations support this hypothesis in the 2D fuse network. We argue that any finite fraction of quenched defects qualitatively reduces the breakdown strength of a wide variety of electrical and mechanical systems in both two and three dimensions. PACS numbers: 77.50. + p, 62.20.Mk, 81.40.Np Breakdown problems are of immense importance in many branches of science and technology. In some problems the medium is dynamic, for example in gas discharge, 1 while in others the medium is, for all practical purposes, static, for example in brittle fracture. 2 In the latter case one may consider any defects in the system to be quenched and the breakdown processes to occur via the growth of cracks from these static seeds. In all real materials there is a finite amount of disorder, and the object of this work is to show that the breakdown properties of real materials can be qualitatively reduced by the presence of a finite fraction of randomly placed quenched defects.The electrical models we consider are displayed in Fig. 1. Figure 1 (a) depicts a model for dielectic breakdown in a random mixture of conductors and insulators. The fraction of conductors (bonds present) is />, and each vacant bond may withstand an electric field strength of 1 V, beyond which it becomes a conductor. Clearly, as the external voltage Kis increased, some of the vacant bonds in the system are exposed to a voltage of more than 1 V. These vacant bonds then break down and become conductors. If enough of the vacant bonds break down the network develops a conducting path and has then suffered dielectric breakdown. Figure Kb) depicts the random fuse network introduced by de Arcangelis, Redner, and Herrmann. 3 Each present bond is a 1-ft fuse that can withstand a current of 1 A (or a voltage of 1 V), beyond which it becomes an insulator. The vacant bonds are insulators that, in their model, never break down to become conductors. As the external voltage is increased, some of the fuses in the system are exposed to more than their rated capacity; they then break down and become insulators. If enough of the fuses are tripped, there is no longer a conducting path across the network and it has suffered breakdown. Two voltages of primary interest in both of these problems are V x , the voltage at which the first bond in the network breaks down, and V b , the voltage at which the network breaks down. Our interest is in the calculation of V x and V b as a function of the fraction p of randomly placed defects. Before discussing the details of the calculations we have performed on these models, we summarize our results and state what we believe to be the most important physical implications of those results.The models display three...

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We show that the geometry of minimum spanning trees (MST) on random graphs is universal. Because of this geometric universality, we are able to characterize the energy of MST using a scaling distribution [P(epsilon)] found using uniform disorder. We show that the MST energy for other disorder distributions is simply related to P(epsilon). We discuss the relationship to invasion percolation, to the directed polymer in a random media, to uniform spanning trees, and also the implications for the broader issue of universality in disordered systems.

We analyze intermittence and roughening of an elastic interface or domain wall pinned in a periodic potential, in the presence of random-bond disorder in 1+1 and 2+1 dimensions. Though the ensemble average behavior is smooth, the typical behavior of a large sample is intermittent, and does not self-average to a smooth behavior. Instead, large fluctuations occur in the mean location of the interface and the onset of interface roughening is via an extensive fluctuation which leads to a jump in the roughness of order lambda, the period of the potential. Analytical arguments based on extreme statistics are given for the number of the minima of the periodicity visited by the interface and for the roughening crossover, which is confirmed by extensive exact ground state calculations.