Varying critical percolation exponents on a multifractal support

TL;DR

This paper investigates how critical percolation exponents vary on a multifractal support, revealing that some exponents remain universal while others depend on local anisotropy, indicating new universality classes.

Contribution

It demonstrates that the correlation length exponent varies with local anisotropy on a multifractal, unlike other exponents that remain consistent with regular lattices.

Findings

$eta$ and $d_f$ exponents match standard percolation values.

The $ u$ exponent varies with local anisotropy and the multifractal's stretching parameter.

Two different $ u$ exponents are identified based on crossing criteria.

Abstract

We study percolation as a critical phenomenon on a multifractal support. The scaling exponents of the the infinite cluster size ($\beta$ exponent) and the fractal dimension of the percolation cluster ($d_f$) are quantities that seem do not depend on local anisotropies. These two quantities have the same value as in the standard percolation in regular bidimensional lattices. On the other side, the scaling of the correlation length ($\nu$ exponent) unfolds new universality classes due to the local anisotropy of the critical percolation cluster. We use two critical exponents $\nu$ according to the percolation criterion for crossing the lattice in either direction or in both directions. Moreover $\nu$ is related to a parameter that characterizes the stretching of the blocks forming the tilling of the multifractal.