Directed Bond Percolation Research Articles

Exact solution for a model on directed lattices

The author defines a model for information propagation through the oriented bonds of a square lattice. The source of information is the origin, and the propagation takes place in only one quadrant, along the diagonal t (time) direction. The information reaches increasing-t layers of lattice points (r, t) according to a probability distribution P(x-xc<<, r, t) dependent only on the previous P(x-xc, r, t-1), where x is the concentration of present bonds. The model shows two distinct behaviours for large values of t: the information will disappear if xxc , or survive forever if xx(inf) c . Taking advantage of the Markovian behaviour and assuming that P is homogeneous, the author gets the values xc=1/2 for the critical concentration, v=1 for the t-correlation length critical exponent, theta =2 for the t/r crossover exponent, and beta =1/2 for the order parameter exponent. Along directions other than t, from the same origin, we get v1=1. The homogeneity assumption is supported by numerical calculations of the time evolution. This evolution is a deterministic cellular automaton, each cell r retaining a real (instead of discrete) value for P. The similarities and differences between the present model and directed bond percolation are also discussed.

A DETERMINANT FOR THE TRANSFER-MATRIX OF THE ONE-DIMENSIONAL DIRECTED BOND PERCOLATION WITH FURTHER NEIGHBOUR BONDS

The rigorous proof of the relation det A(L)=-[∏i=1Lqii](1+q1fL(q1,q2 ⋯, qL)) is given. This relation has been used to obtain the exact critical surface and the correlation length exponents of the one-dimensional directed bond percolation problem with bonds connecting 1~Lth nearest neighbours. A(L) is related to the transfer-matrix by (7). qi is related to the occupation probability of the i-th nearest neighbour bond pi by qi=1–pi (i=1, 2, …, L). fL is a multi-variable polynomial in qi’s.

Finite-size scaling for directed bond percolation with and without cycles on a triangular lattice

The authors apply finite-size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic, directed cyclic and undirected bond percolation on a triangular lattice. The results are in good agreement with known estimates, and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases, as well as the distinction between locally directed and fully isotropic problems. In addition, the authors draw tentative comments on the extension, to directed problems, of the relation A= pi eta (where A is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point), known to hold for various isotropic systems.

On phase diagrams for directed percolation problems

The authors propose a real space renormalisation group scheme for anisotropic directed bond percolation in two dimensions which, differing from previous phenomenological or small-cell real space renormalisation groups, is sensitive to the existence of an infinite cluster along any lattice direction. This enables them to explore new aspects of the phase diagram for the problem, which were not accessible through existing treatments; in particular, the crossover exponent between one- and two-dimensional behaviour is shown to be exactly one for directed percolation. Numerical estimates are in agreement with known inequalities.

Percolation and motion in a simple random environment

The authors present a particularly simple model of deterministic classical motion in a two-dimensional random environment. As the parameters of the model are varied, a transition occurs from all trajectories being localised to some being extended. They construct a mean-field theory for this transition, and relate the model exactly to percolation models in particular parameter ranges. They point out that it is a member of a new class of site percolation analogues of direct bond percolation.

Real-space renormalisation group for anisotropic directed bond percolation in arbitrary dimensions

A real-space renormalisation group transformation for anisotropic directed bond percolation is defined in arbitrary dimensions. For the symmetric case ( phi =0), the percolation threshold pc, obtained is very good in two dimensions, asymptotically exact for large dimensions and quite good in between. The flow diagram is correct for all anisotropies. For the non-symmetric case ( phi =0), the transformation can be generalised and the flow diagram obtained is compatible with the Domany-Kinzel hypothesis (1981).

Directed bond percolation on a square lattice. I. Analytical results

A rigorous mathematical formulation of directed percolation is given in terms of matrix recursion equations. The Domany-Kinzel limit is rederived in a very simple way. Expansion around this limit yields surprisingly good results for the percolation probabilities already to first order.

On square lattice directed percolation and resistance models

Recent papers by Dhar et al. (1981) and Dhar (1982) showed that the standard directed percolation model, with blocked or one-way bonds, is dual to a percolation model with one-way or two-way bonds. This duality is exploited to improve lower bounds for the critical probability of the directed bond percolation model on the square lattice by considering k-order backflows. Two intuitive claims made in the papers mentioned above are verified by the author. Wedge angles of the directed percolation model and the dual model exist almost surely and sum to pi . As the order of backflows k to infinity , the limit of the Dhar-Barma-Phani bounds is the correct critical probability. The proofs use techniques from the theories of subadditive processes and first passage percolation.

Randomly directed bond percolation: a position-space renormalisation group approach

The position-space renormalisation group technique is applied to the randomly directed bond percolation problem. There are three types of bonds in the lattice considered. The bond is undirected with probability u. The bond randomly oriented along one of the two possible directions is present with probability q. The bond is broken with probability 1-u-q. A recursion relation fixed point different from that for the usual undirected bond percolation problem is found. The critical exponents and the percolation threshold probabilities are obtained for percolation in the square lattice.

Directed and diode percolation

We study the novel percolation phenomena that occur in random-lattice networks consisting of resistor-like and diode-like bonds. Resistor bonds connect or "transmit information" in either direction along their length, while diodes connect in one direction only. We first treat the special case of directed bond percolation, in which the diodes are aligned along a preferred axis. Mean-field theory shows that clusters become extremely anisotropic near the percolation transition and that their shapes are characterized by two correlation lengths, one parallel and one transverse to the preferred axis. These lengths diverge with exponents ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=1$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}=\frac{1}{2}$, respectively, from which we can show that the upper critical dimension for this system must be five. We also treat a more general random network on the square lattice containing resistors and diodes of arbitrary orientation. Duality arguments are applied to obtain exact results for the location of phase transitions in this system. We then use a position-space renormalization-group approach to map out the phase diagram and calculate critical exponents. This system has an isotropic percolating phase, and phases which percolate in only one direction. Novel types of transitions occur between these phases, in which the diode orientation plays a fundamental role. These percolating phases meet with the nonpercolating phase along a line of multicritical points, where concentration and orientational fluctuations are simultaneously critical.

Confluent singularities in directed bond percolation

Existing series for directed bond percolation on the square, honeycomb and face centred cubic lattices are analysed with a view to detecting their confluent singularities. The authors compare our results with the confluent singularities in Reggeon field theory, which has been shown to be in the same universality class.

Directed percolation and Reggeon field theory

Directed bond percolation is shown to be in the same universality class as Reggeon field theory. The critical behaviour and critical exponents near the percolation threshold are thereby inferred.

Pair-connectedness for directed bond percolation on some two-dimensional lattices by series methods

The author obtains low-density series expansions of the moments mu n(p)(n=0,2,4) of the pair-connectedness for the acyclic directed square, triangular and honeycomb lattices. Lower bounds to the critical probability of each of the three lattices are also obtained.

Series expansions for the directed-bond percolation problem

Series expansion methods have been used to estimate the critical probability, pc, and to investigate percolation behaviour for the directed-bond problem on most of the usual lattices. High-density series expansions of P(p) and low-density series expansions of S(p) were obtained for the acyclic directed square (SQ), triangular (T), SC, BCC and FCC lattices, and also the cyclic directed triangular lattice. The results suggest that directed models need not fall into the same universality class as the corresponding undirected models. The results for the acyclic and cyclic directed triangular lattices further suggest that different directings of the same lattice may yield models falling into different universality classes.