Multifractal properties of resistor diode percolation

Stenull

2012

Phys. Rev. E

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Long linear polymers in strongly disordered media are well described by self-avoiding walks (SAWs) on percolation clusters and a lot can be learned about the statistics of these polymers by studying the length distribution of SAWs on percolation clusters. This distribution encompasses 2 distinct averages, viz., the average over the conformations of the underlying cluster and the SAW conformations. For the latter average, there are two basic options, one being static and one being kinetic. It is well known for static averaging that if the disorder of the underlying medium is weak, this disorder is redundant in the sense the renormalization group; i.e., differences to the ordered case appear merely in nonuniversal quantities. Using dynamical field theory, we show that the same holds true for kinetic averaging. Our main focus, however, lies on strong disorder, i.e., the medium being close to the percolation point, where disorder is relevant. Employing a field theory for the nonlinear random resistor network in conjunction with a real-world interpretation of the corresponding Feynman diagrams, we calculate the scaling exponents for the shortest, the longest, and the mean or average SAW to 2-loop order. In addition, we calculate to 2-loop order the entire family of multifractal exponents that governs the moments of the the statistical weights of the elementary constituents (bonds or sites of the underlying fractal cluster) contributing to the SAWs. Our RG analysis reveals that kinetic averaging leads to renormalizability whereas static averaging does not, and hence, we argue that the latter does not lead to a well-defined scaling limit. We discuss the possible implications of this finding for experiments and numerical simulations which have produced widespread results for the exponent of the average SAW. To corroborate our results, we also study the well-known Meir-Harris model for SAWs on percolation clusters. We demonstrate that the Meir-Harris model leads back up to 2-loop order to the renormalizable real-world formulation with kinetic averaging if the replica limit is consistently performed at the first possible instant in the course of the calculation.

Section: Observables and Averagesmentioning

confidence: 99%

Linear polymers in disordered media: The shortest, the longest, and the mean self-avoiding walk on percolation clusters

Stenull

2012

Phys. Rev. E

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“…For details we refer to Refs. [7,8] where we derived and analyzed a field theoretic model for the noisy RRDN. The dynamic functional embodying this model near the transitions from the nonpercolating to the directed percolating phases has the same form as the functional (2.2).…”

Section: The Noisy Rrdnmentioning

confidence: 99%

Transport properties of directed percolation clusters at the upper critical dimension

Stenull

J. Phys.: Condens. Matter

2005

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We study the transport properties of directed percolation clusters at the upper critical dimension dc = 4 + 1, where critical fluctuations induce logarithmic corrections to the leading (mean-field) scaling behavior. Employing field theory and renormalization group methods we calculate these logarithmic corrections up to and including the next to leading correction for a variety of observables, viz. the connectivity, i.e., the probability that two given points are connected, the average twopoint resistance and some of the fractal masses describing percolation clusters. Furthermore, we study logarithmic corrections for the multifractal moments of the current distribution on directed percolation clusters.

“…We obtain 36) where the second sum runs over all lattice vectors b i between site i and its nearest neighbors.…”

Section: B Field Theoretic Hamiltonianmentioning

confidence: 99%

Percolating granular superconductors

Stenull²

2003

Phys. Rev. E

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We investigate diamagnetic fluctuations in percolating granular superconductors. Granular superconductors are known to have a rich phase diagram including normal, superconducting and spin glass phases. Focusing on the normal-superconducting and the normal-spin glass transition at low temperatures, we study he diamagnetic susceptibility χ (1) and the mean square fluctuations of the total magnetic moment χ (2) of large clusters. Our work is based on a random Josephson network model that we analyze with the powerful methods of renormalized field theory. We investigate the structural properties of the Feynman diagrams contributing to the renormalization of χ (1) and χ (2) . This allows us to determine the critical behavior of χ (1) and χ (2) to arbitrary order in perturbation theory.