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 ppc, the squares and triangles are spatially well mixed and the point configurations belong to the hyperuniform class III with the exponent 0<<1. This means the existence of the hyperuniform-antihyperuniform transition at p=pc. 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.