Hyperuniformity of self-similar point processes

Abstract

We study hyperuniformity of self-similar point processes arising from substitution rules in two dimensions. In particular, we derive a sufficient condition for hyperuniformity of these point processes only in terms of the associated substitution matrix. This condition applies to a wide class of examples for which hyperuniformity had not yet been established, including most well-known examples of planar self-similar tilings. In particular, we show that the Godrèche-Lançon-Billard substitution rule gives rise to hyperuniform point processes with singular continuous diffraction. Furthermore, we prove that hyperuniformity is not an MLD invariant, contradicting standing conjectures.