The Susceptible-Infected-Recovered (SIR) model is studied in multilayer networks with arbitrary number of links across the layers. By following the mapping to bond percolation we give the exact expression for the epidemic threshold and the fraction of the infected individuals in arbitrary number of layers. The case of a multilayer network formed by two interconnected networks is specifically st...

I discuss a recently unveiled feature in the dynamics of two dimensional coarsening systems on the lattice with Ising symmetry: they first approach a critical percolating state via the growth of a new length scale, and only later enter the usual dynamic scaling regime. The time needed to reach the critical percolating state diverges with the system size. These observations are common to Glauber...

We consider the nearest-neighbor simple random walk on Z, d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy > 0 exceeds the threshold for bond percolation on Z. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the 2n-step ...

Two related issues are explored for bond percolation on Z (with d ≥ 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point pfin satisfies pfin ≥ pc, and strict inequality is proved when either d is sufficiently large, or d ≥ 7 and the model is sufficiently spr...