Spectral dimensionality of random superconducting networks

Abstract

We compute the spectral dimensionality d̃ of random superconducting-normal networks by directly examining the low-frequency density of states at the percolation threshold. We find that d̃=4.1±0.2 and 5.8±0.3 in two and three dimensions, respectively, which confirms the scaling relation d̃=2d/(2-s/ν), where s is the superconducting exponent and ν the correlation-length exponent for percolation. We also consider the one-dimensional problem where scaling arguments predict, and our numerical simulations confirm, that d̃=0. A simple argument provides an expression for the density of states of the localized high-frequency modes in this special case. We comment on the connection between our calculations and the ‘‘termite’’ problem of a random walker on a random superconducting-normal network and point out difficulties in inferring d̃ from simulations of the termite problem.

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