  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.
