Truncation of long-range percolation model with square non-summable interactions

We consider some problems related to the truncation question in long-range percolation. It is given probabilities that certain long-range oriented bonds are open; assuming that this probabilities are not summable, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. This question is still open if the set of vertices is $\Z^2$. We give some conditions in which the answer is affirmative. One of these results generalize the previous result in [Alves, Hil\'ario, de Lima, Valesin, Journ. Stat. Phys. {\bf 122}, 972 (2017)]

Truncated long-range percolation on oriented graphs

We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are open; assuming that the sum of these probabilities is infinite, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. We give some conditions in which the answer is affirmative. We also translate some of our results on oriented percolation to the context of a long-range contact process.Comment: 9 pages, 1 figur

An integrable modification of the critical Chalker-Coddington network model

We consider the Chalker-Coddington network model for the Integer Quantum Hall Effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models, with two loop flavours. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra, and parameterised by the loop fugacity $n$. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-Gas theory, whereas the other two branches (3,4) couple the two loop flavours, and relate to an $SU(2)_r \times SU(2)_r / SU(2)_{2r}$ Wess-Zumino-Witten (WZW) coset model for the particular values $n= -2\cos[\pi/(r+2)]$ where $r$ is a positive integer. The truncated Chalker-Coddington model is the $n=0$ point of branch 4. By numerical diagonalisation, we find that its universality class is neither an analytic continuation of the WZW coset, nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.Comment: 34 pages, 18 figures, 3 appendice

Long-range contact process and percolation on a random lattice

We study the phase transition phenomena for long-range oriented percolation and contact process. We studied a contact process in which the range of each vertex are independent, updated dynamically and given by some distribution $N$. We also study an analogous oriented percolation model on the hyper-cubic lattice, here there is a special direction where long-range oriented bonds are allowed; the range of all vertices are given by an i.i.d. sequence of random variables with common distribution $N$. For both models, we prove some results about the existence of a phase transition in terms of the distribution $N$

Critical point network for drainage between rough surfaces

In this paper, we present a network method for computing two-phase flows between two rough surfaces with significant contact areas. Low-capillary number drainage is investigated here since one-phase flows have been previously investigated in other contributions. An invasion percolation algorithm is presented for modeling slow displacement of a wetting fluid by a non wetting one between two rough surfaces. Short-correlated Gaussian process is used to model random rough surfaces.The algorithm is based on a network description of the fracture aperture field. The network is constructed from the identification of critical points (saddles and maxima) of the aperture field. The invasion potential is determined from examining drainage process in a flat mini-channel. A direct comparison between numerical prediction and experimental visualizations on an identical geometry has been performed for one realization of an artificial fracture with a moderate fractional contact area of about 0.3. A good agreement is found between predictions and observations

Effective Dielectric Tensor for Electromagnetic Wave Propagation in Random Media

We derive exact strong-contrast expansions for the effective dielectric tensor \epeff of electromagnetic waves propagating in a two-phase composite random medium with isotropic components explicitly in terms of certain integrals over the $n$-point correlation functions of the medium. Our focus is the long-wavelength regime, i.e., when the wavelength is much larger than the scale of inhomogeneities in the medium. Lower-order truncations of these expansions lead to approximations for the effective dielectric constant that depend upon whether the medium is below or above the percolation threshold. In particular, we apply two- and three-point approximations for \epeff to a variety of different three-dimensional model microstructures, including dispersions of hard spheres, hard oriented spheroids and fully penetrable spheres as well as Debye random media, the random checkerboard, and power-law-correlated materials. We demonstrate the importance of employing $n$-point correlation functions of order higher than two for high dielectric-phase-contrast ratio. We show that disorder in the microstructure results in an imaginary component of the effective dielectric tensor that is directly related to the {\it coarseness} of the composite, i.e., local volume-fraction fluctuations for infinitely large windows. The source of this imaginary component is the attenuation of the coherent homogenized wave due to scattering. We also remark on whether there is such attenuation in the case of a two-phase medium with a quasiperiodic structure.Comment: 40 pages, 13 figure

Universality classes in nonequilibrium lattice systems

This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.Comment: Updated comprehensive review, 62 pages (two column), 29 figs included. Scheduled for publication in Reviews of Modern Physics in April 200