Universality class of site and bond percolation on multifractal scale-free planar stochastic lattice.

Abstract

In this article, we investigate both site and bond percolation on a weighted planar stochastic lattice (WPSL), which is a multifractal and whose dual is a scale-free network. The characteristic property of percolation is that it exhibits threshold phenomena as we find sudden or abrupt jump in spanning probability across p_{c} accompanied by the divergence of some other observable quantities, which is reminiscent of a continuous phase transition. Indeed, percolation is characterized by the critical behavior of percolation strength P(p)∼(p_{c}-p)^{β}, mean cluster size S∼(p_{c}-p)^{-γ}, and the system size L∼(p_{c}-p)^{-ν}, which are known as the equivalent counterpart of the order parameter, susceptibility, and correlation length, respectively. Moreover, the cluster size distribution function n_{s}(p_{c})∼s^{-τ} and the mass-length relation M∼L^{d_{f}} of the spanning cluster also provide useful characterization of the percolation process. We numerically obtain a value for p_{c} and for all the exponents such as β,ν,γ,τ, and d_{f}. We find that, except for p_{c}, all the exponents are exactly the same in both bond and site percolation despite the significant difference in the definition of cluster and other quantities. Our results suggest that the percolation on WPSL belongs to a new universality class, as its exponents do not share the same value as for all the existing planar lattices. Besides, like all other cases, its site and bond type belong to the same universality class.