Critical exponents for isotropically directed percolation on hierarchical lattices.

Abstract

We use renormalization-group and Monte Carlo calculations to investigate isotropically directed percolation on hierarchical lattices. We describe a general renormalization-group approach to exactly determine the critical exponents for these lattices. This approach is equivalent to, albeit simpler than, the method of cluster generating functions and can be applied to various hierarchical lattices. In the case of isotropically directed percolation, we need to determine two critical exponents, namely, β_{scc} and β_{out} associated, respectively, with the giant strongly connected component and the giant out component in the neighborhood of the percolation threshold. Our method is applied to different constructions of the hierarchical lattice, in particular, to a small-world hierarchical lattice which includes long-range bonds and exhibits a form of explosive percolation.