Aftermath epidemics: Percolation on the sites visited by generalized random walks.

We study percolation on the sites of a finite lattice visited by a generalized random walk of finite length with periodic boundary conditions. More precisely, consider Levy flights and walks with finite jumps of length >1 [like Knight's move random walks (RWs) in two dimensions and generalized Knight's move RWs in 3D]. In these walks, the visited sites do not form (as in ordinary RWs) a single connected cluster, and thus percolation on them is nontrivial. The model essentially mimics the spreading of an epidemic in a population weakened by the passage of some devastating agent-like diseases in the wake of a passing army or of a hurricane. Using the density of visited sites (or the number of steps in the walk) as a control parameter, we find a true continuous percolation transition in all cases except for the 2D Knight's move RWs and Levy flights with Levy parameter σ≥2. For 3D generalized Knight's move RWs, the model is in the universality class of pacman percolation, and all critical exponents seem to be simple rationals, in particular, ß=1. For 2D Levy flights with 0<σ<2, scale invariance is broken even at the critical point, which leads at least to very large corrections in finite-size scaling, and even very large simulations were unable to unambiguously determine the critical exponents.

Universality class of epidemic percolation transitions driven by random walks.

Inspired by the recent viral epidemic outbreak and its consequent worldwide pandemic, we devise a model to capture the dynamics and the universality of the spread of such infectious diseases. The transition from a precritical to the postcritical phase is modeled by a percolation problem driven by random walks on a two-dimensional lattice with an extra average number ρ of nonlocal links per site. Using finite-size scaling analysis, we find that the effective exponents of the percolation transitions as well as the corresponding time thresholds, extrapolated to the infinite system size, are ρ dependent. We argue that the ρ dependence of our estimated exponents represents a crossover-type behavior caused by the finite-size effects between the two limiting regimes of the system. We also find that the universal scaling functions governing the critical behavior in every single realization of the model can be well described by the theory of extreme values for the maximum jumps in the order parameter and by the central limit theorem for the transition threshold.