Real space renormalization group theory of the percolation model

A cluster expansion renormalization group method in real space is-developed to determine the critical properties of the percolation model. In contrast to previous renormalization group approaches, this method considers the cluster size distribution (free energy) rather than the site or bond probability distribution (coupling constants) and satisfies the basic renormalization group requirement of free energy conservation. In the construction of the renormalization group transformation, new couplings are generated which alter the topological structure of the clusters and which must be introduced in the original system. Predicted values of the critical exponents appear to converge to presumed exact values as higher orders in the expansion are considered. The method can in principle be extended to different lattice structures, as well as to different dimensions of space.This paper is dedicated to Prof. Philippe Choquard.     

Correlated percolation — a real-space renormalization group study

A quenched-correlated percolation system is studied by using the real-space renormalization group method. We chose the nearest-neighbour short-range order α as the correlation parameter. By treating the correlated probability self-consistently in terms of site occupation probability p and α, we found that, in two dimensions, there is only one nontrivial physical fixed point at random percolation and the correlation is an irrelevant parameter which always leads to the same universality.     

Statistics of directed self-avoiding walks on percolation clusters at the percolation threshold

We here study directed self-avoiding walks on site diluted square lattice at the percolation threshold by two parameter real space renormalization group method. We found \(v_\parallel ^{p_c } = 1.00\) and \(v_ \bot ^{p_c } = 0.4348\) from cell-to-cell transformation method. This \(v_ \bot ^{p_c } \) value is then compared with the modified Alexander-Orbach formula that \(v_ \bot ^{p_c } = {{d_S } \mathord{\left/ {\vphantom {{d_S } {2d_L }}} \right. \kern-0em} {2d_L }}\) whered _s is the fracton dimension andd _L is the spreading dimension of the infinite directed percolation cluster.     

Histogram Monte Carlo position-space renormalization group: Applications to the site percolation

We study site percolation on the square lattice and show that, when augmented with histogram Monte Carlo simulations for large lattices, the cell-to-cell renormalization group approach can be used to determine the critical probability accurately. Unlike the cell-to-site method and an alternate renormalization group approach proposed recently by Sahimi and Rassamdana, both of which rely onab initio numerical inputs, the cell-to-cell scheme is free of prior knowledge and thus can be applied more widely.     

Monte Carlo renormalization group study of the percolation problem of discs with a distribution of radii

A real space renormalization group is formulated for continuum (off-lattice) percolation problems. It is applied to the system of overlapping discs with a variety of distributions of disc radii. Monte Carlo method is used for obtaining recursion relations. The results support universality: The Harris criterion seems to work for percolation. The position of the critical point shows stability against introducing a distribution in the disc radii.Supported in part by SFB 125 Aachen-Jülich-Köln     

Heisenberg ferromagnet with Dzyaloshinsky-Moriya interactions: A real space renormalization group approach

We study the anisotropic Heisenberg ferromagnet with anti-symmetric Dzyaloshinsky-Moriya (DM) interactions for arbitrary dimensions. We use a real space renormalization group approach of the Migdal-Kadanoff type in order to obtain the phase diagrams and critical exponents. The effect of the Dzyaloshinsky-Moriya term in the global phase diagram is worked out in detail. Our results suggest that in all dimensions the effect of the DM interaction is to renormalize the parameters of the anisotropic exchange Hamiltonian. Finally we discuss the modification of hyperscaling associated with the zero temperature Heisenberg-like fixed point.     

Thermal phase transitions at the percolation threshold

We present a simple argument, valid in all dimensions for a wide range of quenched dilute spin models, for the equality of the pair correlation function and the pair connectedness function at the point p = p_c, T = 0 is approached along the T axis.     

Random walks on bond percolation clusters: ac hopping conductivity below the threshold

The frequency dependence of the ac hopping conductivity in two and three dimensional lattices with random interruptions is calculated by Monte Carlo simulation of random walks on bond percolation clusters. At low frequencies the real and imaginary parts of the ac conductivity vanish linearly and quadratically with the frequency, respectively. The critical behaviour of the imaginary part of the ac conductivity below the percolation threshold is shown to depend on the long time limit of the mean square displacement of random walksR ² , while the real part depends on the time constant of the system as well.R ² is found to diverge with an exponentu=2- according to the conjecture of Stauffer.     

Numerical study of currents fluctuations in metal-dielectric composites at the percolation threshold

In this paper, dc current distribution through a 2D square metal-dielectric network is determined and analyzed near the percolation threshold. The current in each link is calculated by solving Kirchhoff’s equations. Using the difference between dielectric and metallic distributions, and the scaling analysis of the peak of large currents, it is found that this peak corresponds to backbone. Furthermore, by examining the critical exponent of current fluctuations, it is found that these fluctuations correspond to the magnetic susceptibility of the network. This correspondence is explained in terms of the fluctuation-dissipation theorem.     

Higher-order calculations in the real space dynamic renormalization group: One dimension

The application of real space dynamic renormalization group methods to the one-dimensional kinetic Ising model, discussed in an earlier paper, is extended to one order higher in perturbation theory than was done previously. It is shown that the treatment of short-range, local quantities is improved in going to higher order in the perturbation expansion, while that of the long-range properties remains largely unaffected. Arbitrariness in the real space mapping function and how it may be exploited to our advantage is duscussed. It is shown that the renormalized Hamiltonian continues to be characterized by one coupling through second order. We find that the single spin-flip kinetic Ising model generates at second-order new spin-flip mechanisms in the renormalized dynamical operator but that their effects are small (at most 2%) over the entire temperature range.     

Solvable real space renormalization group for the kinetic Ising chain

A real-space renormalization group for the one-dimensional kinetic Ising model is established. The parameter space of the model must be enlarged to include non-Markovian kernels in the equation of motion. The recursion relations for these kernels can be iterated analytically so that the global flow under the renormalization group can be traced exactly. The resulting fixed-point equation is non-Markovian.     

Quantum spins and quasiperiodicity: a real space renormalization group approach

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.     

Different self-avoiding walks on percolation clusters: A small-cell real-space renormalization-group study

We calculate the average number of stepsN for edge-to-edge, normal, and indefinitely growing self-avoiding walks (SAWs) on two-dimensional critical percolation clusters, using the real-space renormalization-group approach, with small H cells. Our results are of the formN=AL ^D _SAW+B, whereL is the end-to-end distance. Similarly to several deterministic fractals, the fractal dimensionsD _SAW for these three different kinds of SAWs are found to be equal, and the differences between them appear in the amplitudesA and in the correction termsB. This behavior is atributed to the hierarchical nature of the critical percolation cluster.     

A renormalization group study of the Anderson Hamiltonian for arbitrary band-filling

The Anderson hamiltonian has been studied for arbitrarily filled bands using the real space renormalization group (RSRG) method. The ground state energy is determined in one dimension for the scale factors n_s = 3 and n_s = 5. Direct application of RSRG gives, for a certain n_s, the results for only n_s number of discrete cases of band-filling. We then devise an interpolation scheme which spans the entire region of the band and gives the ground state energy for any general filling. The results, as provided by our scheme, satisfy the desired particle-hole symmetry exactly and are always an upper bound to the exact one. The non-linear dependence of the band energy on its filling is represented correctly, even for small n_s. Finally, we discuss how the results improve for large n_s.     

Real space renormalization group investigation of three-dimensional Ising spin glasses

Using real space renormalization group techniques we determine the phase diagram of bond dilute frustrated nearest-neighbor Ising three-dimensional simple cubic (sc) and body-centered cubic (bcc) systems.