Absence of the nonpercolating phase for percolation on the nonplanar Hanoi network

Abstract

We investigate bond percolation on the nonplanar Hanoi network (HN-NP), which was studied previously [Boettcher et al. Phys. Rev. E 80, 041115 (2009)]. We calculate the fractal exponent of a subgraph of the HN-NP, which gives a lower bound for the fractal exponent of the original graph. This lower bound leads to the conclusion that the original system does not have a nonpercolating phase, where only finite-size clusters exist for $p>0$, or equivalently, that the system exhibits either the critical phase, where infinitely many infinite clusters exist, or the percolating phase, where a unique giant component exists. Monte Carlo simulations support our conjecture.