Recursive Percolation

TL;DR

This paper introduces recursive percolation, a new lattice model built on critical percolation clusters, revealing a hierarchy of universality classes with unique critical exponents in two and three dimensions.

Contribution

It presents a novel recursive percolation model, determines thresholds up to five generations in 2D, and uncovers new universality classes with distinct critical exponents.

Findings

Percolation thresholds determined up to n=5 in 2D.

Critical clusters become more compact with increasing n.

New universality classes with unique exponents are identified.

Abstract

We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and show that it can be repeated recursively any number $n$ of generations. In two dimensions, we determine the percolation thresholds up to $n=5$. The corresponding critical clusters become more and more compact as $n$ increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any $n$. This family of exponents differs from any previously known universality class, and cannot be accommodated by existing analytical methods. We confirm that recursive percolation is well defined also in three dimensions.