Daniel Roca González - Hyperuniformity of hat tilings - Hatfest скачать в хорошем качестве

Daniel Roca González - Hyperuniformity of hat tilings - Hatfest

Hyperuniformity (of Torquato-Stillinger) and number rigidity (of Ghosh-Peres) are two long range order properties of point processes (or tilings), which have attracted a lot of attention by mathematical physicists and probabilists alike due to their manifold connections to material science, chemistry and biology. While these properties are by now fairly well-understood for one-dimensional tilings, very little was known about them for higher-dimensional tilings until recently. In this talk we establish hyperuniformity for many tilings coming from inflation rules, including hat tilings. For other tilings, such as Penrose tilings and certain tilings associated to the hat, we are also able to prove number rigidity, making them be among the first examples with finite local complexity, hyperuniformity and number rigidity in higher dimensions. Our proof of these facts uses three main ingredients: Spectral characterizations of hyperuniformity and number rigidity, spherical diffraction for the group of motions of the Euclidean plane (also known as powder diffraction) and self-similarity equations for spherical diffraction which generalize self-similarity equations of Baake and Grimm for ordinary diffraction. Our method applies more generally to substitution tilings of arbitrary dimensions for which there is a sufficiently large eigenvalue gap in the reduced substitution matrix. A talk from Hatfest at the Mathematical Institute, University of Oxford. See https://sites.google.com/view/thegrimmnetw... for more information.