The Resistance of Feynman Diagrams and the Percolation Backbone Dimension

We present a new view of Feynman diagrams for the field theory of transport on percolation clusters. The diagrams for random resistor networks 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 $D_B = 2 + \epsilon /21 - 172\epsilon^2 /9261 + 2 \epsilon^3 (- 74639 + 22680 \zeta (3))/4084101$, where $\epsilon = 6-d$ with $d$ being the spatial dimension and $\zeta (3) = 1.202057..$.Comment: 10 pages, 2 figure

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 non-percolating 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.Comment: 21 pages, 5 figures, to appear in Phys. Rev.

Transport on Directed Percolation Clusters

We study random lattice networks consisting of resistor like and diode like 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 $\phi$ 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.Comment: 4 page

Fresh look at randomly branched polymers

We develop a new, dynamical field theory of isotropic randomly branched polymers, and we use this model in conjunction with the renormalization group (RG) to study several prominent problems in the physics of these polymers. Our model provides an alternative vantage point to understand the swollen phase via dimensional reduction. We reveal a hidden Becchi-Rouet-Stora (BRS) symmetry of the model that describes the collapse ($\theta$-)transition to compact polymer-conformations, and calculate the critical exponents to 2-loop order. It turns out that the long-standing 1-loop results for these exponents are not entirely correct. A runaway of the RG flow indicates that the so-called $\theta^\prime$-transition could be a fluctuation induced first order transition.Comment: 4 page

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 \phi_{\mathcal S \mathnormal} and \phi_{\mathcal S \mathnormal}^\infty for the special and the ordinary transition, respectively, to one-loop order.Comment: 19 pages, 3 figure

Master Operators Govern Multifractality in Percolation

Using renormalization group methods we study multifractality in percolation at the instance of noisy random resistor networks. We introduce the concept of master operators. The multifractal moments of the current distribution (which are proportional to the noise cumulants $C_R^{(l)} (x, x^\prime)$ of the resistance between two sites x and $x^\prime$ located on the same cluster) are related to such master operators. The scaling behavior of the multifractal moments is governed exclusively by the master operators, even though a myriad of servant operators is involved in the renormalization procedure. We calculate the family of multifractal exponents ${\psi_l}$ for the scaling behavior of the noise cumulants, $C_R^{(l)} (x, x^\prime) \sim | x - x^\prime |^{\psi_l /\nu}$, where $\nu$ is the correlation length exponent for percolation, to two-loop order.Comment: 6 page

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 $\phi$. Upon blending the $\epsilon$-expansion result with the exact value $\phi =1$ for one dimension by a rational approximation, we obtain for two dimensions $\phi = 1.29\pm 0.05$. This value is in agreement with the recent simulations of a two-dimensional random diode network by Inui, Kakuno, Tretyakov, Komatsu, and Kameoka, 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, T\"{a}uber, and Schwabl and clear up some minor shortcomings.Comment: 24 pages, 2 figure

Field theory of directed percolation with long-range spreading

It is well established that the phase transition between survival and extinction in spreading models with short-range interactions is generically associated with the directed percolation (DP) universality class. In many realistic spreading processes, however, interactions are long ranged and well described by L\'{e}vy-flights, i.e., by a probability distribution that decays in $d$ dimensions with distance $r$ as $r^{-d-\sigma}$. We employ the powerful methods of renormalized field theory to study DP with such long range, L\'{e}vy-flight spreading in some depth. Our results unambiguously corroborate earlier findings that there are four renormalization group fixed points corresponding to, respectively, short-range Gaussian, L\'{e}vy Gaussian, short-range DP and L\'{e}vy DP, and that there are four lines in the $(\sigma, d)$ plane which separate the stability regions of these fixed points. When the stability line between short-range DP and L\'{e}vy DP is crossed, all critical exponents change continuously. We calculate the exponents describing L\'{e}vy DP to second order in $\epsilon$-expansion, and we compare our analytical results to the results of existing numerical simulations. Furthermore, we calculate the leading logarithmic corrections for several dynamical observables.Comment: 12 pages, 3 figure

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, $V\sim I^r$. 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 $d_{\scriptsize red} = 1/\nu$ at least to order {\sl 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, d_{\scriptsize min} = 2 - \epsilon /6 - [ 937/588 + 45/49 (\ln 2 -9/10 \ln 3)] (\epsilon /6)^2 + {\sl O} (\epsilon^3) verifies a previous calculation by one of us. For the backbone dimension we find D_B = 2 + \epsilon /21 - 172 \epsilon^2 /9261 + 2 (- 74639 + 22680 \zeta (3))\epsilon^3 /4084101 + {\sl O} (\epsilon^4), where $\zeta (3) = 1.202057...$, in agreement to second order in $\epsilon$ with a two-loop calculation by Harris and Lubensky.Comment: 29 pages, 7 figure

Critical Exponents for Diluted Resistor Networks

An approach by Stephen 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. By a decomposition of the principal Feynman diagrams we obtain a type of diagrams which again can be interpreted as resistor networks. This new 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 $\phi=1+\epsilon /42 +4\epsilon^2 /3087$ verifies a previous calculation by Lubensky and Wang, which itself was based on the Potts--model formulation of the random resistor network.Comment: 27 pages, 14 figure