Patterns in Random Tilings

The purpose of this project is to develop new techniques and explore new territories for determinantal point processes coming from random tilings of planar domains.

In the past two decades great progress has been made on the understanding of the remarkable patterns that random tilings of planar domains exhibit. Yet, many models are still out of reach with state-of-the-art techniques and several conjectures remain unsolved.

Special attention will be devoted to random tilings models where the randomness is induced from certain doubly periodic weights on the underlying bipartite graph.

By very recent progress by the applicant and collaborators, new techniques have been developed that may help understanding the limiting fluctuations of such models when the domains get large. More precisely, the proposal has the following objectives:1.

Develop methods for asymptotic studies of the correlation function for random tilings of large domains, including measures from doubly periodic weights.2.

Derive new asymptotic formulas for matrix-valued orthogonal polynomials, in particular those that are associated with doubly periodic tilings.3.

Formulate and investigate natural extensions of Schur processes that include doubly periodic weights that have a special integrable structure.4.

Study the  fluctuations of the height function on macroscopic and mesocopic scales,  classify the universal Gaussian log-correlated fields and esablish their universality.