Percolation of systems having hyperuniformity or giant number-fluctuations

We generate point configurations (PCs) by thresholding the local energy of the Ashkin-Teller model in two dimensions (2D) and study the percolation transition at different values of $\lambda$ along the critical Baxter line by varying the threshold that controls the particle density $\rho$. For all values of $\lambda$, the PCs exhibit power-law correlations with a decay exponent $a$ that remains independent of $\rho$ and varies continuously with $\lambda$. For $\lambda < 0$, where the PCs are hyperuniform, the percolation critical behavior is identical to that of ordinary percolation. In contrast, for $\lambda > 0$, the configurations exhibit giant number fluctuations, and all critical exponents vary continuously, but form a superuniversality class of percolation transition in 2D.