Hyperuniform properties of square-triangle tilings

We study hyperuniform properties for the square-triangle tilings. The tiling is generated by a local growth rule, where squares or triangles are iteratively attached to its boundary. The introduction of the probability $p$ in the growth rule, which controls the expansion of square and triangle domains, enables us to obtain various square-triangle random tilings systematically. We analyze the degree of the regularity of the point configurations, which are defined as the vertices on the square-triangle tilings, in terms of hyperuniformity. It is clarified that for $p<{p}_{c}\phantom{\rule{0.28em}{0ex}}({p}_{c}\ensuremath{\sim}0.5)$, the system can be regarded as a phase separation between square and triangular lattice domains and the variance of the point configurations obeys the scaling law ${\ensuremath{\sigma}}^{2}\ensuremath{\sim}O({R}^{2\ensuremath{-}\ensuremath{\alpha}})$ with $\ensuremath{\alpha}<0$. The configurations are antihyperuniform. On the other hand, for $p>{p}_{c}$, the squares and triangles are spatially well mixed and the point configurations belong to the hyperuniform class III with the exponent $0<\ensuremath{\alpha}<1$. This means the existence of the hyperuniform-antihyperuniform transition at $p={p}_{c}$. We also examine the structure factor of the square-triangle tilings. It is clarified that the peak structures in the large-wave-number regime are mostly common to all square-triangle tilings, while those in the small-wave-number regime strongly depend on whether the point configurations are hyperuniform or antihyperuniform.