Directed Percolation and the Golden Ratio

Applying the theory of Yang-Lee zeros to nonequilibrium critical phenomena, we investigate the properties of a directed bond percolation process for a complex percolation parameter p. It is shown that for the Golden Ratio p = (1± √ 5)/2 and for p = 2 the survival probability of a cluster can be computed exactly. PACS numbers: 02.50.-r,64.60.Ak,05.50.+q Directed percolation (DP) is an anisotropic variant of ordinary percolation in which activity can only percolate along a given direction in space. Regarding this direction as a temporal degree of freedom, DP can be interpreted as a dynamical process. Directed percolation represents one of the most prominent universality classes of nonequilibrium phase transitions from a fluctuating active phase into a non-fluctuating absorbing state [1, 2, 3, 4]. A simple realization of DP is directed bond percolation. In this model the bonds of a tilted square lattice are conducting with probability p and non-conducting with probability 1 − p (see Figure 1). The order parameter which characterizes the phase transition is the probability P (∞) that a randomly chosen site belongs to an infinite cluster. A cluster consists of all sites that are connected by a directed path of conducting bonds to the sites that generate the cluster at time t = 0. For p > pc this probability is finite whereas it vanishes for p ≤ pc. Close to the phase transition P (∞) is known to vanish algebraically as P (∞) ∼ (p − pc). Although DP can be defined and simulated easily, it is one of the very few systems for which – even in one spatial dimension – no analytical solution is known, suggesting that DP is a non-integrable process. In fact, the values of the percolation threshold and the critical exponents are not simple numerical fractions but seem to be irrational instead. Currently the best estimates for directed bond percolation in 1+1 dimensions are pc = 0.6447001(1) and β = 0.27649(4) [5]. ‡ To whom correspondence should be addressed ([email protected]) Directed Percolation and the Golden Ratio 2