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Bhattacharjee S, Ramola K. Green's functions for random resistor networks. Phys Rev E 2023; 108:044148. [PMID: 37978714 DOI: 10.1103/physreve.108.044148] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2022] [Accepted: 06/26/2023] [Indexed: 11/19/2023]
Abstract
We analyze random resistor networks through a study of lattice Green's functions in arbitrary dimensions. We develop a systematic disorder perturbation expansion to describe the weak disorder regime of such a system. We use this formulation to compute ensemble-averaged nodal voltages and bond currents in a hierarchical fashion. We verify the validity of this expansion with direct numerical simulations of a square lattice with resistances at each bond exponentially distributed. Additionally, we construct a formalism to recursively obtain the exact Green's functions for finitely many disordered bonds. We provide explicit expressions for lattices with up to four disordered bonds, which can be used to predict nodal voltage distributions for arbitrarily large disorder strengths. Finally, we introduce a novel order parameter that measures the overlap between the bond current and the optimal path (the path of least resistance), for a given resistance configuration, which helps to characterize the weak and strong disorder regimes of the system.
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Affiliation(s)
- Sayak Bhattacharjee
- Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India
| | - Kabir Ramola
- Tata Institute of Fundamental Research, Hyderabad 500107, India
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Janssen HK, Stenull O. Linear polymers in disordered media: the shortest, the longest, and the mean self-avoiding walk on percolation clusters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:011123. [PMID: 22400528 DOI: 10.1103/physreve.85.011123] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2011] [Indexed: 05/31/2023]
Abstract
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.
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Affiliation(s)
- Hans-Karl Janssen
- Institut für Theoretische Physik III, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
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Affiliation(s)
- A. B. Harris
- a Raymond and Beverly Sackler Faculty of Exact Sciences , School of Physics and Astronomy, Tel Aviv University , Ramat Aviv, Tel Aviv , 69978 , Israel
- b Department of Physics , University of Pennsylvania , Philadelphia , Pennsylvania , 19104 , U.S.A
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Stenull O. Multifractality in a broad class of disordered systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:015101. [PMID: 15324110 DOI: 10.1103/physreve.70.015101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2004] [Indexed: 05/24/2023]
Abstract
We study multifractality in a broad class of disordered systems which includes, e.g., the diluted x-y model. Using renormalized field theory we analyze the scaling behavior of cumulant averaged dynamical variables (in case of the x-y model the angles specifying the directions of the spins) at the percolation threshold. Each of the cumulants has its own independent critical exponent, i.e., there are infinitely many critical exponents involved in the problem. Working out the connection to the random resistor network, we determine these multifractal exponents to two-loop order. Depending on the specifics of the Hamiltonian of each individual model, the amplitudes of the higher cumulants can vanish and in this case, effectively, only some of the multifractal exponents are required.
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Affiliation(s)
- Olaf Stenull
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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Janssen HK, Stenull O. Corrections to scaling in random resistor networks and diluted continuous spin models near the percolation threshold. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:026118. [PMID: 14995531 DOI: 10.1103/physreve.69.026118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2003] [Indexed: 05/24/2023]
Abstract
We investigate corrections to scaling induced by irrelevant operators in randomly diluted systems near the percolation threshold. The specific systems that we consider are the random resistor network and a class of continuous spin systems, such as the x-y model. We focus on a family of least irrelevant operators and determine the corrections to scaling that originate from this family. Our field theoretic analysis carefully takes into account that irrelevant operators mix under renormalization. It turns out that long standing results on corrections to scaling are respectively incorrect (random resistor networks) or incomplete (continuous spin systems).
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Affiliation(s)
- Hans-Karl Janssen
- Institut für Theoretische Physik III, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
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Stenull O, Janssen HK. Logarithmic corrections to scaling in critical percolation and random resistor networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:036129. [PMID: 14524854 DOI: 10.1103/physreve.68.036129] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2003] [Indexed: 05/24/2023]
Abstract
We study the critical behavior of various geometrical and transport properties of percolation in six dimensions. By employing field theory and renormalization group methods we analyze fluctuation induced logarithmic corrections to scaling up to and including the next-to-leading order correction. Our study comprehends the percolation correlation function, i.e., the probability that two given points are connected, and some of the fractal masses describing percolation clusters. To be specific, we calculate the mass of the backbone, the red bonds, and the shortest path. Moreover, we study key transport properties of percolation as represented by the random resistor network. We investigate the average two-point resistance as well as the entire family of multifractal moments of the current distribution.
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Affiliation(s)
- Olaf Stenull
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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Hinrichsen H, Stenull O, Janssen HK. Multifractal current distribution in random-diode networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:045104. [PMID: 12005904 DOI: 10.1103/physreve.65.045104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2001] [Indexed: 05/23/2023]
Abstract
Recently it has been shown analytically that electric currents in a random-diode network are distributed in a multifractal manner [O. Stenull and H. K. Janssen, Europhys. Lett. 55, 691 (2001)]. In the present paper we investigate the multifractal properties of a random diode network at the critical point by numerical simulations. We analyze the currents running on a directed percolation cluster and confirm the field-theoretic predictions for the scaling behavior of moments of the current distribution. It is pointed out that a random diode network is a particularly good candidate for a possible experimental realization of directed percolation.
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Affiliation(s)
- Haye Hinrichsen
- Theoretische Physik, Fachbereich 8, Universität GH Wuppertal, 42097 Wuppertal, Germany
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Stenull O, Janssen HK. Multifractal properties of resistor diode percolation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:036124. [PMID: 11909182 DOI: 10.1103/physreve.65.036124] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2001] [Indexed: 05/23/2023]
Abstract
Focusing on multifractal properties we investigate electric transport on random resistor diode networks at the phase transition between the nonpercolating and the directed percolating phase. Building on first principles such as symmetries and relevance we derive a field theoretic Hamiltonian. Based on this Hamiltonian we determine the multifractal moments of the current distribution that are governed by a family of critical exponents [psi(l)]. We calculate the family [psi(l)] to two-loop order in a diagrammatic perturbation calculation augmented by renormalization group methods.
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Affiliation(s)
- Olaf Stenull
- Institut für Theoretische Physik III, Heinrich-Heine-Universität, Universitätsstrasse 1, 40225 Düsseldorf, Germany
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Stenull O, Janssen HK, Oerding K. Effects of surfaces on resistor percolation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:056128. [PMID: 11414982 DOI: 10.1103/physreve.63.056128] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2000] [Indexed: 05/23/2023]
Abstract
We study the effects of surfaces on resistor percolation at the instance of a semi-infinite geometry. Particularly we are interested in the average resistance between two connected ports located on the surface. Based on general grounds as symmetries and relevance we introduce a field theoretic Hamiltonian for semi-infinite random resistor networks. We show that the surface contributes to the average resistance only in terms of corrections to scaling. These corrections are governed by surface resistance exponents. We carry out renormalization-group improved perturbation calculations for the special and the ordinary transition. We calculate the surface resistance exponents phiS and phiS(infinity) for the special and the ordinary transition, respectively, to one-loop order.
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Affiliation(s)
- O Stenull
- Institut für Theoretische Physik III, Heinrich-Heine-Universität, Universitätsstrasse 1, 40225 Düsseldorf, Germany
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Stenull O, Janssen HK. Noisy random resistor networks: renormalized field theory for the multifractal moments of the current distribution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:036103. [PMID: 11308705 DOI: 10.1103/physreve.63.036103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2000] [Indexed: 05/23/2023]
Abstract
We study the multifractal moments of the current distribution in randomly diluted resistor networks near the percolation threshold. When an external current is applied between two terminals x and x(') of the network, the lth multifractal moment scales as M((l))(I)(x,x(')) approximately equal /x-x'/(psi(l)/nu), where nu is the correlation length exponent of the isotropic percolation universality class. By applying our concept of master operators [Europhys. Lett. 51, 539 (2000)] we calculate the family of multifractal exponents [psi(l)] for l>or=0 to two-loop order. We find that our result is in good agreement with numerical data for three dimensions.
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Affiliation(s)
- O Stenull
- Institut für Theoretische Physik III, Heinrich-Heine-Universität, Düsseldorf, Germany
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Pennetta C, Trefan G, Reggiani L. Scaling law of resistance fluctuations in stationary random resistor networks. PHYSICAL REVIEW LETTERS 2000; 85:5238-5241. [PMID: 11102230 DOI: 10.1103/physrevlett.85.5238] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2000] [Indexed: 05/23/2023]
Abstract
In a random resistor network we consider the simultaneous evolution of two competing random processes consisting in breaking and recovering the elementary resistors with probabilities W(D) and W(R). The condition W(R)>W(D)/(1+W(D)) leads to a stationary state, while in the opposite case, the broken resistor fraction reaches the percolation threshold p(c). We study the resistance noise of this system under stationary conditions by Monte Carlo simulations. The variance of resistance fluctuations <deltaR2> is found to follow a scaling law |p-p(c)|(-kappa(0)) with kappa(0) = 5.5. The proposed model relates quantitatively the defectiveness of a disordered media with its electrical and excess-noise characteristics.
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Affiliation(s)
- C Pennetta
- Dipartimento di Ingegneria dell'Innovazione e Istituto Nazionale di Fisica della Materia, Universita di Lecce, Via Arnesano, I-73100 Lecce, Italy
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Janssen HK, Stenull O. Diluted networks of nonlinear resistors and fractal dimensions of percolation clusters. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:4821-4834. [PMID: 11031523 DOI: 10.1103/physreve.61.4821] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/1999] [Indexed: 05/23/2023]
Abstract
We study random networks of nonlinear resistors, which obey a generalized Ohm's law V approximately Ir. Our renormalized field theory, which thrives on an interpretation of the involved Feynman diagrams as being resistor networks themselves, is presented in detail. By considering distinct values of the nonlinearity r, we calculate several fractal dimensions characterizing percolation clusters. For the dimension associated with the red bonds we show that dred = 1/nu at least to order O(epsilon 4), with nu being the correlation length exponent, and epsilon = 6 - d, where d denotes the spatial dimension. This result agrees with a rigorous one by Coniglio. Our result for the chemical distance, dmin = 2 - epsilon/6 - [937/588 + 45/49(ln 2 - 9/10 ln 3)](epsilon/6)2 + O(epsilon 3) verifies a previous calculation by one of us. For the backbone dimension we find DB = 2 + epsilon/21 - 172 epsilon 2/9261 + 2[-74639 + 22680 zeta(3)]epsilon 3/4084101 + O(epsilon 4), where zeta(3) = 1.202057..., in agreement to second order in epsilon with a two-loop calculation by Harris and Lubensky.
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Affiliation(s)
- H K Janssen
- Institut für Theoretische Physik III, Heinrich-Heine-Universität, Düsseldorf, Germany
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Stenull O, Janssen HK, Oerding K. Critical exponents for diluted resistor networks. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 59:4919-30. [PMID: 11969444 DOI: 10.1103/physreve.59.4919] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/1998] [Indexed: 04/18/2023]
Abstract
An approach by Stephen [Phys. Rev. B 17, 4444 (1978)] is used to investigate the critical properties of randomly diluted resistor networks near the percolation threshold by means of renormalized field theory. We reformulate an existing field theory by Harris and Lubensky [Phys. Rev. B 35, 6964 (1987)]. By a decomposition of the principal Feynman diagrams, we obtain diagrams which again can be interpreted as resistor networks. This interpretation provides for an alternative way of evaluating the Feynman diagrams for random resistor networks. We calculate the resistance crossover exponent phi up to second order in epsilon=6-d, where d is the spatial dimension. Our result phi=1+epsilon/42+4epsilon(2)/3087 verifies a previous calculation by Lubensky and Wang, which itself was based on the Potts-model formulation of the random resistor network.
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Affiliation(s)
- O Stenull
- Institut für Theoretische Physik III, Heinrich-Heine-Universität, Universitätsstrasse 1, 40225 Düsseldorf, Germany
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Kahng B, Batrouni GG, Redner S. Logarithmic voltage anomalies in random resistor networks. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/20/13/004] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Fourcade B, Tremblay A. Field theory and second renormalization group for multifractals in percolation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:4095-4104. [PMID: 9963120 DOI: 10.1103/physreve.51.4095] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Deutsch JM, Zacher RA. Probability distribution for a multifractal field. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:R8-R10. [PMID: 9961296 DOI: 10.1103/physreve.49.r8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Goldschmidt YY, Blum T. Directed polymers in a random medium: Universal scaling behavior of the probability distribution. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:R2979-R2982. [PMID: 9960450 DOI: 10.1103/physreve.47.r2979] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Adler J, Aharony A, Blumenfeld R, Harris AB, Meir Y. Distribution of the logarithms of currents in percolating resistor networks. II. Series expansions. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:5770-5782. [PMID: 10004523 DOI: 10.1103/physrevb.47.5770] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Aharony A, Blumenfeld R, Harris AB. Distribution of the logarithms of currents in percolating resistor networks. I. Theory. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:5756-5769. [PMID: 10004522 DOI: 10.1103/physrevb.47.5756] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Tremblay RR, Albinet G, Tremblay A. Noise and crossover exponents in conductor-insulator mixtures and superconductor-conductor mixtures. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 45:755-767. [PMID: 10001116 DOI: 10.1103/physrevb.45.755] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Duplantier B, Ludwig AW. Multifractals, operator-product expansion, and field theory. PHYSICAL REVIEW LETTERS 1991; 66:247-251. [PMID: 10043758 DOI: 10.1103/physrevlett.66.247] [Citation(s) in RCA: 38] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Bunde A, Havlin S, Roman HE. Multifractal features of random walks on random fractals. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:6274-6277. [PMID: 9903921 DOI: 10.1103/physreva.42.6274] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kahng B. Negative moments of current distribution in random resistor networks. PHYSICAL REVIEW LETTERS 1990; 64:914-917. [PMID: 10042113 DOI: 10.1103/physrevlett.64.914] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Aharony A, Blumenfeld R, Breton P, Fourcade B, Harris AB, Meir Y, Tremblay A. Negative moments of currents in percolating resistor networks. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 40:7318-7320. [PMID: 9991130 DOI: 10.1103/physrevb.40.7318] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Wang J, Harris AB. Cohn's theorem for elastic networks. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 40:7272-7278. [PMID: 9991116 DOI: 10.1103/physrevb.40.7272] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Harris AB. Scaling of the negative moments of the harmonic measure in diffusion-limited aggregates. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 39:7292-7294. [PMID: 9947390 DOI: 10.1103/physrevb.39.7292] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Sahimi M. Statistical physics of linear and nonlinear, scalar vector transport processes in disordered media. ACTA ACUST UNITED AC 1988. [DOI: 10.1016/0920-5632(88)90041-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Kahng B, Batrouni GG, Redner S, Herrmann HJ. Electrical breakdown in a fuse network with random, continuously distributed breaking strengths. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 37:7625-7637. [PMID: 9944059 DOI: 10.1103/physrevb.37.7625] [Citation(s) in RCA: 120] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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29
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Balberg I, Wagner N, Hearn DW, Ventura JA. Computer study of the electrical noise in high-dimensional percolating systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 37:3829-3831. [PMID: 9945012 DOI: 10.1103/physrevb.37.3829] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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30
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Fourcade B, Breton P, Tremblay A. Multifractals and critical phenomena in percolating networks: Fixed point, gap scaling, and universality. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 36:8925-8928. [PMID: 9942748 DOI: 10.1103/physrevb.36.8925] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Coniglio A, Redner S. Multifractal structure of the incipient infinite percolating cluster. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 36:5631-5634. [PMID: 9942219 DOI: 10.1103/physrevb.36.5631] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Harris AB. Field-theoretic formulation of the randomly diluted nonlinear resistor network. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 35:5056-5065. [PMID: 9940689 DOI: 10.1103/physrevb.35.5056] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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