Fuse model on a randomly diluted hierarchical lattice

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Abstract
The first breaking current of the randomly diluted fuse network in a hierarchical diamond lattice is studied theoretically and numerically. The authors show that the prediction proposed by Duxbury et al. (1987) for Euclidean lattices should be applicable, and that it is correct: the average current necessary to break the first bond of the lattice decreases on average as the inverse of the logarithm of the system size. Its validity is shown to rely solely on the self-averaging of the conductivity with system size. Due to the peculiar geometry of the lattice, the breaking current as a function of the lattice size exhibits a series of plateaux followed by sudden variations between them. Once the first few bonds are broken, the rest of the breaking of the network is no longer comparable to that of a Euclidean lattice: the mean breaking current for the entire lattice decreases with system size much slower than the current necessary to break the first bond.