Percolation on multifractal, scale-free weighted planar stochastic porous lattice

Abstract

We introduce the Weighted Planar Stochastic Porous Lattice (WPSPL), a geometrically disordered substrate generated by iteratively subdividing a unit square. At each step a block is selected with probability proportional to its area, divided into four parts, and one sub-block is retained (removed) with probability $q$ ($1-q$). We show analytically that the WPSPL exhibits multifractality for each of its infinitely many nontrivial conserved quantities and demonstrate numerically that its snapshots at different times are statistically self-similar. The dual of the lattice forms a complex network with a power-law degree distribution. Motivated by these properties of this porous lattice, we study bond percolation on the WPSPL, determine the percolation threshold, and estimate the critical exponents $\alpha$, $\beta$, and $\gamma$ associated with the specific heat, order parameter, and susceptibility, respectively. The exponents vary continuously with $q$, reflecting a family of distinct universality classes as the global dimension of the lattice depends on $q$. Remarkably, the Rushbrooke inequality, $\alpha + 2\beta + \gamma \ge 2$, is satisfied in near equality. Notably, the nonporous case ($q=1$) has a global dimension $2$ but lies outside the universality class of conventional two-dimensional lattices. Our results highlight how geometric disorder, multifractality, scale-free coordination number disorder, and porosity produce unconventional critical behavior.