Chase-escape percolation on the 2D square lattice. (English) Zbl 1528.92040
Summary: 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 forever with non-zero probability, if \(p > p_c\) with \(p_c < 1\). Earlier simulations showed that \(p_c\) is very close to 1/2. Using Monte Carlo simulations in \(D = 2\), we estimate the value of \(p_c\) to be \(0.49451 \pm 0.00001\) and the critical exponents are consistent with the undirected percolation universality class. We check that at \(p_c\), the correlation functions at large length scales are rotationally invariant. We define a discrete-time parallel-update version of the model, which brings out the relation between chase-escape and undirected bond percolation. We further show that for all \(p < p_c\), in \(D\) dimensions, the probability that the number of predators in the absorbing configuration is greater than \(s\) is bounded from below by \(\exp(-Kp^{-1} s^{1/D})\), where \(K\) is some \(p\)-independent constant. This is in contrast to the exponentially decaying cluster size distribution in the standard percolation theory. Even so, the scaling function for the cluster size distribution for \(p\) near \(p_c\) decays exponentially: the stretched exponential behavior dominates for \(s \gg s^\ast\), but \(s^\ast\) diverges near \(p_c\). We also study the problem starting from an 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 \(p_c < p < 1\), 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. This is caused by the prey sites at the leading edge being eaten up by predators. The fluctuations of the front follow KPZ scaling both above and below the depinning transition at \(p = 1\).
MSC:
| 92D30 | Epidemiology |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
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