Self-dual quasiperiodic 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.