Critical exponents for diluted resistor networks

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

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.

Distribution functions in percolation problems

Percolation clusters are random fractals whose geometrical and transport properties can be characterized with the help of probability distribution functions. Using renormalized field theory, we determine the asymptotic form of various such distribution functions in the limits where certain scaling variables become small or large. Our study includes the pair-connection probability, the distributions of the fractal masses of the backbone, the red bonds, and the shortest, the longest, and the average self-avoiding walk between any two points on a cluster, as well as the distribution of the total resistance in the random resistor network. Our analysis draws solely on general, structural features of the underlying diagrammatic perturbation theory, and hence our main results are valid to arbitrary loop order.

Scaling behavior of linear polymers in disordered media

It has long been known that the universal scaling properties of linear polymers in disordered media are well described by the statistics of self-avoiding walks (SAWs) on percolation clusters and their critical exponent nu(SAW), with the SAW implicitly referring to the average SAW. Hitherto, static averaging has been commonly used, e.g., in numerical simulations, to determine what the average SAW is. We assert that only kinetic, rather than static, averaging can lead to asymptotic scaling behavior and corroborate our assertion by heuristic arguments and a renormalizable field theory. Moreover, we calculate to two-loop order nu(SAW), the exponent nu(max) for the longest SAW, and a family of multifractal exponents nu(alpha).

Corrections to scaling in random resistor networks and diluted continuous spin models near the percolation threshold

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).

Logarithmic corrections to scaling in critical percolation and random resistor networks

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.

Percolating granular superconductors

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 the diamagnetic susceptibility chi((1)) and the mean square fluctuations of the total magnetic moment chi((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 chi((1)) and chi((2)). This allows us to determine the critical behavior of chi((1)) and chi((2)) to arbitrary order in perturbation theory.

Normal stresses at the gelation transition

A simple Rouse-type model, generalized to incorporate the effects of chemical cross-links, is used to obtain a theoretical prediction for the critical behavior of the normal-stress coefficients Psi(1) and Psi(2) in polymeric liquids when approaching the gelation transition from the sol side. While the exact calculation shows Psi(2) identical with 0, a typical result for these types of models, an additional scaling ansatz is used to demonstrate that Psi(1) diverges with a critical exponent l=k+z. Here, k denotes the critical exponent of the shear viscosity and z the exponent governing the divergence of the time scale in the Kohlrausch decay of the shear-stress relaxation function. For cross-links distributed according to mean-field percolation, this scaling relation yields l=3, in accordance with an exact expression for the first normal-stress coefficient based on a replica calculation. Alternatively, using three-dimensional percolation for the cross-link ensemble we find the value l approximately 4.9. Results on time-dependent normal-stress response are also presented.

Multifractal properties of resistor diode percolation

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.

Conductivity of continuum percolating systems

We study the conductivity of a class of disordered continuum systems represented by the Swiss-cheese model, where the conducting medium is the space between randomly placed spherical holes, near the percolation threshold. This model can be mapped onto a bond percolation model where the conductance sigma of randomly occupied bonds is drawn from a probability distribution of the form sigma(-a). Employing the methods of renormalized field theory we show to arbitrary order in epsilon expansion that the critical conductivity exponent of the Swiss-cheese model is given by t(SC)(a) = (d-2)nu + max[phi,(1-a)(-1)], where d is the spatial dimension and nu and phi denote the critical exponents for the percolation correlation length and resistance, respectively. Our result confirms a conjecture that is based on the "nodes, links, and blobs" picture of percolation clusters.

Relevance of percolation theory to the vulcanization transition

The relationship between vulcanization and percolation is explored from the perspective of renormalized local field theory. We show to arbitrary order in perturbation theory that the vulcanization and percolation correlation functions are governed by the same Gell-Mann-Low renormalization-group equation. Hence, all scaling aspects of the vulcanization transition are reigned by the critical exponents of the percolation universality class.

Nonlinear random resistor diode networks and fractal dimensions of directed percolation clusters

We study nonlinear random resistor diode networks at the transition from the nonpercolating to the directed percolating phase. The resistor-like bonds and the diode-like bonds under forward bias voltage obey a generalized Ohm's law V approximately I(r). Based on general grounds such as symmetries and relevance we develop a field theoretic model. We focus on the average two-port resistance, which is governed at the transition by the resistance exponent straight phi(r). By employing renormalization group methods we calculate straight phi(r) for arbitrary r to one-loop order. Then we address the fractal dimensions characterizing directed percolation clusters. Via considering distinct values of the nonlinearity r, we determine the dimension of the red bonds, the chemical path, and the backbone to two-loop order.

Effects of surfaces on resistor percolation

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.

Noisy random resistor networks: renormalized field theory for the multifractal moments of the current distribution

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.

Transport on directed percolation clusters

We study random lattice networks consisting of resistorlike and diodelike bonds. For investigating the transport properties of these random resistor diode networks we introduce a field-theoretic Hamiltonian amenable to renormalization group analysis. We focus on the average two-port resistance at the transition from the nonpercolating to the directed percolating phase and calculate the corresponding resistance exponent straight phi to two-loop order. Moreover, we determine the backbone dimension D(B) of directed percolation clusters to two-loop order. We obtain a scaling relation for D(B) that is in agreement with well known scaling arguments.

Critical dynamics of gelation

Shear relaxation and dynamic density fluctuations are studied within a Rouse model, generalized to include the effects of permanent random crosslinks. We derive an exact correspondence between the static shear viscosity and the resistance of a random resistor network. This relation allows us to compute the static shear viscosity exactly for uncorrelated crosslinks. For more general percolation models, which are amenable to a scaling description, it yields the scaling relation k=straight phi-beta for the critical exponent of the shear viscosity. Here beta is the thermal exponent for the gel fraction, and straight phi is the crossover exponent of the resistor network. The results on the shear viscosity are also used in deriving upper and lower bounds on the incoherent scattering function in the long-time limit, thereby corroborating previous results.

Random resistor-diode networks and the crossover from isotropic to directed percolation

By employing the methods of renormalized field theory, we show that the percolation behavior of random resistor-diode networks near the multicritical line belongs to the universality class of isotropic percolation. We construct a mesoscopic model from the general epidemic process by including a relevant isotropy-breaking perturbation. We present a two-loop calculation of the crossover exponent straight phi. Upon blending the varepsilon-expansion result with the exact value straight phi=1 for one dimension by a rational approximation, we obtain straight phi=1.29+/-0.05 for two dimensions. This value is in agreement with the recent simulations of a two-dimensional random diode network by Inui, et al. [Phys. Rev. E 59, 6513 (1999)], who found an order parameter exponent beta different from those of isotropic and directed percolation. Furthermore, we reconsider the theory of the full crossover from isotropic to directed percolation by Frey, Tauber, and Schwabl [Europhys. Lett. 26, 413 (1994); Phys. Rev. E 49, 5058 (1994)], and clear up some minor shortcomings.

Diluted networks of nonlinear resistors and fractal dimensions of percolation clusters

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.

Resistance of Feynman diagrams and the percolation backbone dimension

We present an alternative view of Feynman diagrams for the field theory of random resistor networks, in which the diagrams are interpreted as being resistor networks themselves. This simplifies the field theory considerably as we demonstrate by calculating the fractal dimension D(B) of the percolation backbone to three loop order. Using renormalization group methods we obtain D(B)=2+epsilon/21-172epsilon(2)/9261+2epsilon(3)[-74 639+22 680zeta(3)]/4 084 101, where epsilon=6-d with d being the spatial dimension and zeta(3)=1.202 057... .