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

TL;DR

This study analyzes site and bond percolation on a multi-multifractal scale-free lattice, revealing a new universality class with identical critical exponents for both types despite structural differences.

Contribution

It provides exact critical thresholds and exponents for percolation on a complex stochastic lattice, establishing a new universality class distinct from known planar lattices.

Findings

Exact percolation threshold p_c obtained.

Critical exponents β, ν, γ, τ, d_f determined.

Site and bond percolation share the same universality class.

Abstract

In this article, we investigate both site and bond percolation on a weighted planar stochastic lattice (WPSL) which is a multi-multifractal and whose dual is a scale-free network. The characteristic properties 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 continuous phase transition. Indeed, percolation is characterized by the critical behavior of percolation strength $P(p)\sim (p_c-p)^\beta$, mean cluster size $S\sim (p_c-p)^{-\gamma}$ and the system size $L\sim (p_c-p)^{-\nu}$ which are known as the equivalent counterpart of the order parameter, susceptibility and correlation length respectively. Moreover, the cluster size distribution function $n_s(p_c)\sim s^{-\tau}$ and the mass-length relation $M\sim L^{d_f}$ of the spanning cluster also provide useful characterization of the percolation process. We obtain an exact value for $p_c$ and for all the exponents such as $\beta, \nu, \gamma, \tau$ and $d_f$. We find that, except $p_c$, 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 and like other cases its site and bond belong to the same universality class.