Fig. 1: Domany–Kinzel model of the directed percolation.
From: Bistability and time crystals in long-ranged directed percolation
a The probability pi,t of site i to be occupied at time t depends on the occupation of its nearest-neighbours \(i\pm \frac{1}{2}\) at time t − 1 and can take discrete values 0, q1 and q2. b Flowchart representation of the DK model. The initial occupation probability is uniform pi,t = 1 = p1. At time t, each site i is either occupied (si,t = 1) or empty (si,t = 0) with probability pi,t and 1 − pi,t, respectively. Time is advanced and local densities \({\{{n}_{i,t}\}}_{i}\) are computed for each site i as averages of the nearest-neighbour occupations at previous time, and these densities determine the occupation probabilities for the next generation, see Eq. (3). The generations at all subsequent times are obtained by iteration. c, d The density n at late times can be used to discern the active and inactive phases, in which n(t = 103) >0 and ≈0, respectively. The dashed lines serve as a reference to locate the phase boundary and are the same for initial densities p1 = 1 (c) and p1 = 0.01 (d). The insets show representative single instances of the DP for the points in the (q1, q2) plane marked with a cross. Here L = 100 and R = 103.
