Lipschitz cutset for fractal graphs and applications to the spread of infections

We consider the fractal Sierpiński gasket or carpet graph in dimension d≥2, denoted by G. At time 0, we place a Poisson point process of particles onto the graph and let them perform independent simple random walks, which in this setting exhibit sub-diffusive behaviour. We generalise the concept of particle process dependent Lipschitz percolation to the (coarse graining of the) space-time graph G×R, where the opened/closed state of space-time cells is measurable with respect to the particle process inside the cell. We then provide an application of this generalised framework and prove the following: if particles can spread an infection when they share a site of G, and if they recover independently at some rate γ>0, then if γ is sufficiently small, the infection started with a single infected particle survives indefinitely with positive probability.