An interacting random walk model produces Turing-like filamentation and a new phase transition in time

We show a new percolation-like phase transition in time, viz., sudden disappearance of long-range spatial connectivity, in a system of interacting random walkers on a square lattice. Our model consists of walkers that are placed randomly or uniformly at sites of a square lattice at time t=0. The dynamics of the walk is simulated by a simple interaction model. At time t, let n be the number of walkers on a lattice site and m the sum of the number of walkers on the neighboring sites. If n≥m the walker hops to one of the neighboring sites with equal probability and if n<m it does a lazy walk, i.e., hops only with probability exp(m-n). We declare a site active if there is at least one walker on it and inactive otherwise. Two neighboring sites have an edge if both are active. We compute the wrapping probability of the clusters formed at each time step, averaging over many simulations. We observe a sharp phase transition in the wrapping probability at a time threshold. A numerical finite-size scaling analysis shows the universality class to be the same as that of standard site percolation. In addition, we observe Turing-like patterns in the clusters as we go through the phase transition, much like laser filamentation in a strong pulse propagating through a self-Kerr ionizing medium.