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

TL;DR

This paper introduces a multifractal, scale-free weighted planar stochastic porous lattice and studies its percolation properties, revealing continuous variation in critical exponents and unconventional critical behavior due to geometric disorder and porosity.

Contribution

The paper presents a novel stochastic porous lattice model with multifractality and analyzes its percolation thresholds and critical exponents, uncovering new universality classes.

Findings

Percolation threshold determined for the lattice.

Critical exponents vary continuously with parameter q.

Unconventional critical behavior due to geometric disorder and porosity.

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.