Self-dual quasiperiodic percolation

TL;DR

This paper investigates two self-dual quasiperiodic models of square-lattice bond percolation, revealing that removing randomness alters critical behavior, with unique fractal dimensions and critical exponents distinct from random percolation.

Contribution

The study introduces two novel quasiperiodic percolation models and characterizes their critical properties, showing they do not belong to the same universality class as random percolation.

Findings

Critical points exhibit discrete scale invariance.

Fractal dimensions differ significantly from random percolation.

Critical exponents are lower than those of random percolation.

Abstract

How does the percolation transition behave in the absence of quenched randomness? To address this question, we study two nonrandom self-dual quasiperiodic models of square-lattice bond percolation. In both models, the critical point has emergent discrete scale invariance, but none of the additional emergent conformal symmetry of critical random percolation. From the discrete sequences of critical clusters, we find fractal dimensions of $D_f=1.911943(1)$ and $D_f=1.707234(40)$ for the two models, significantly different from $D_f=91/48=1.89583...$ of random percolation. The critical exponents $\nu$, determined through a numerical study of cluster sizes and wrapping probabilities on a torus, are also well below the $\nu=4/3$ of random percolation. While these new models do not appear to belong to a universality class, they demonstrate how the removal of randomness can fundamentally change the critical behavior.