  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 (RW) in 2 dimensions and generalized
knight's move RW in 3d). In these walks, the visited sites do not form (as in
ordinary RW) a single connected cluster, and thus percolation on them is
non-trivial. 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 2-d knight's
move RW and Levy flights with Levy parameter $\sigma \geq 2$. For 3-d
generalized knight's move RW, the model is in the universality class of Pacman
percolation, and all critical exponents seem to be simple rationals, in
particular $\beta=1$. For 2-d Levy flights with $0 <\sigma < 2$, 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 determine unambiguously the critical exponents.
