2020/10/31 by Aanjaneya Kumar, Peter Grassberger, Deepak Dhar
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Biology #Bounded function #Combinatorics #Complex Network Analysis Techniques #Critical exponent #Directed percolation #Exponential decay #Exponential distribution #Exponential function #Exponential growth #Geometry #Mathematical analysis #Mathematics #Monte Carlo method #Opinion Dynamics and Social Influence #Percolation (cognitive psychology) #Percolation theory #Percolation threshold #Physics #Quantum mechanics #Scaling #Square lattice #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Topology (electrical circuits) #cond-mat.stat-mech #math.PR #q-bio.PE
paper · pdf · doi:10.1016/j.physa.2021.126072
Substantially revised with added figures and results
arxiv created 2021/02/12 · openalex publication_date 2021/05/03 · arxiv updated 2021/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate p, and predator particles spread only to neighboring sites occupied by prey particles at rate 1, killing the prey particle that existed at that site. It was found that the prey can survive with non-zero probability, if p>pc with pc<1. Using Monte Carlo simulations on the square lattice, we estimate the value of pc = 0.49451 ± 0.00001, and the critical exponents are consistent with the undirected percolation universality class. We define a discrete-time parallel-update version of the model, which brings out the relation between chase-escape and undirected bond percolation. For all p < pc in D-dimensions, the number of predators in the absorbing configuration has a stretched-exponential distribution in contrast to the exponential distribution in the standard percolation theory. We also study the problem starting from the line initial condition with predator particles on all lattice points of the line y=0 and prey particles on the line y=1. In this case, for pc<p < 1, the center of mass of the fluctuating prey and predator fronts travel at the same speed. This speed is strictly smaller than the speed of an Eden front with the same value of p, but with no predators. At p=1, the fronts undergo a depinning transition. The fluctuations of the front follow Kardar-Parisi-Zhang scaling both above and below this depinning transition.