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Directed Percolation and Directed Animals 1 Directed Percolation and Directed Animals Deepak Dhar Theoretical Physics Group Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai 400 005, (INDIA) Abstract These lectures provide an introduction to the directed percolation and directed animals problems, from a physicist’s point of view. The probabilistic cellular automaton formulation of directed percolation is introduced. The planar duality of the diode-resistor-insulator percolation problem in two dimensions, and relation of the directed percolation to undirected first passage percolation problem are described. Equivalence of the d-dimensional directed animals problem to (d − 1)-dimensional Yang-Lee edge-singularity problem is established. Self-organized critical formulation of the percolation problem, which does not involve any fine-tuning of coupling constants to get critical behavior is briefly discussed. I. DIRECTED PERCOLATION Although directed percolation (DP ) was defined by Broadbent and Hammersley along with undirected percolation in their first paper in 1957 [1], it did not receive much attention in percolation theory until late seventies, when Blease obtained rather precise estimates of some critical exponents [2] using the fact that one can efficiently generate very long series expansions for this problem on the computer. Since then, it has been studied extensively, both as a simple model of stochastic processes, and because of its applications in different physical situations as diverse as star formation in galaxies [3], reaction-diffusion systems [4], conduction in strong electric field in semiconductors [5], and biological evolution [6]. The large number of papers devoted to DP in current physics literature are due to the fact that the critical behavior of a very large number of stochastic evolving systems are in the DP universality class. In this lecture, I briefly sketch some of these connections. Additional references may be found in [7-9], or the book by Stauffer and Aharony [10]. A. The Forest-Fire Model Can we embed the conventional bond percolation, say on a square lattice, in a stochastic process, which evolves randomly in time? This may be done as follows: we think of the lattice as a forest, with each site occupied by a tree. A tree can be in one of three states: green, burning or ash. We assume that time t = 0, there is a single burning tree at the origin, and all other trees are green. The time evolution is discrete, and governed by the following rules: arXiv:1703.07541v1 [cond-mat.stat-mech] 22 Mar 2017 (i) A burning tree at time t, becomes ash at time (t + 1). (ii) A green tree not neighbouring any burning neighbours remains green. If it has r burning neighbours at time t, it catches fire and becomes a burning tree with probability (1 − qr), with q =1 − p. (iii) Ash remains ash for all subsequent times. Here p is a parameter 0 <p< 1. If p is small, eventually the fire dies, and a configuration of only green trees and ash results which does not change further. It is easy to that the probability that the final configuration of burnt trees is a given set S, is the same as the probability that the cluster connected to the origin is S in the bond percolation problem. There is a non zero probability that fire survives infinitely long for p > 1/2, and this exactly equals the probability of the origin belonging to the infinite cluster. 2 All questions relating to statistics of percolation clusters can be reformulated as questions about the probabilities of different sink states of the model. The advantage of this formulation is that it suggests other questions related to the time evolution of the system, e.g. what is the velocity with which the fire front spreads outwards for p>pc? B. Forest-Fire Model with Regrowth A somewhat different model results if we allow the burnt trees to regrow, and become green again after some time-lag. The main qualitative difference over the previous case is that now the configuration with all sites green is the unique sink-state of the system. It turns out that the precise duration of the time-lag is not very important, and it is just as well if we assume that burnt sites become green at the next time step. This allows us to work with only two states per site: burning and green. We replace the rule (i) in the previous model by (i′) the burning site becomes green at the next time step and we do not need rule (iii) any more. Alternatively, this may be thought of as a model of infection of disease in a population, where the infected individual usually recovers after a period of illness. This is the simplest formulation of the DP problem. It is interesting that this new model has a non-trivial phase transition even in 1-dimension. It has been studied a lot, and very precise estimates are available for critical parameters and critical exponents [11], but an exact solution has not been possible so far. C. The Probabilistic Cellular Automaton Model Another terminology which is useful for the previous model is that of a probabilistic cellular automaton [12]. At each site i, we have discrete variable n(i,t) at time t, taking values 0 or 1, (0 = no fire, 1 = fire). Evolution is discrete time, parallel and local. In the simplest case, n(i,t + 1) depends only on the value of [n(i − 1,t)+ n(i +1,t)]. Let pr be the probability that n(i,t+1) is 1 if n(i−1,t)+n(i+1,t)= r. We assume p0 = 0. This model with two parameter (p1,p2) is known as the Domany-Kinzel model [13]. For p1 = p2 = p, this model is the directed site- percolation 2 problem, and for p1 = p, p2 =2p − p it corresponds to the directed bond- percolation at concentration p. Domany and Kinzel showed that this model can be solved exactly for p2 = 1, when the problem reduces to that of annihilating random walkers on a line. D. Relation to Quantum Spin Chains The directed percolation process is basically a Markov process that involves allowing for the possibility of growth, propagation or death of some activity (here fire). In some applications, it is useful to think of this in continous time. Then, we say n(i,t) = 1, it can change to 0 with rate 1, if n(i,t) = 0, and n(i − 1,t)+ n(i +1,t) = r then it can become 1 with rate rλ. The master equation for the evolution of Prob(C,t), the probability that configuration of the system is C at time t, which has the form d ′ Prob(C,t)= W ′ Prob(C ,t), (1) dt CC XC′ can be thought of as a Schrodinger equation for the evolution of the system. The ‘wavefunction’ to be in configuration C at time t is Prob(C,t), and the Hamiltonian is the matrix WCC′ . One can then use the experience and insight gained from the study of quantum Hamiltonians such as of spin-chains to learn about stochastic evolving systems. This has been a very useful approach in recent years (see [14] for a review), though this approach has a long history (see, for example [15]). In this particular case, if we think of ni = 1 as spin up, and ni = 0 as spin down, then W becomes the Hamiltonian + of a linear chain of spin-1/2 particles, with nearest neighbour couplings. Introducing ai and ai as the Pauli spin raising and lowering operators at site i, we see that the Hamiltonian W can be written in the form + + + + + + W = ai + λai ai ai+1 + ai−1 − ai ai 1+ λai+1ai+1 + λai−1ai−1 . (2) Xi Note that W is not hermitian. This quantum mechanical formalism can be developed further into quantum field theoretical formulation. This was historically the first technique used to estimate critical exponents in all dimensions using the ǫ-expansion techniques (for references, see [9]). We shall not discuss these further here. 3 E. Scaling Theory for Directed Percolation The general theoretical treatment of the undirected percolation (existence of critical threshold, exponential decay below criticality etc.) goes through unchanged for the directed percolation problem. There are two major differences: there is no unique infinite percolation cluster. The infinite cluster depends on the choice of the origin. Secondly, below but near the critical threshold pc, the clusters are large, but anisotropic. We have to define two different correlation lengths ξk and ξ⊥, which determine the average size of cluster along the preferred direction, and transverse to it. Near −νk −ν⊥ pc, these lengths diverge as (pc − p) and (pc − p) , where νk and ν⊥ are different exponents with νk >ν⊥. The probability that the site (R⊥, Rk) is connected to the origin, when the concentration is p = pc + ǫ defines the green’s function G(R⊥, Rk,ǫ). For small ǫ and large R⊥ and Rk , this function is expected to reduce to a scaling function of two arguments −β/νk 1/ν⊥ 1/νk G(R⊥, Rk,ǫ) ≈ Rk g(ǫR⊥ , ǫRk ) (3) νk β The function g(x1, x2) is expected to decrease as exp(−|x2| ) for x2 → −∞, and increase as x2 for x → +∞. For a fixed ǫ> 0, the percolation occurs within a cone of angle θ(p), and this angle varies as ǫνk−ν⊥ for small ǫ> 0. Other exponents such as γ which characterizes the divergence of mean cluster size near pc can be expressed in terms of these three exponents β,ν⊥ and νk by using scaling relations.
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