Patterns in Random Tilings

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. The general purpose of this project is develop new techniques for solving such conjectures and explore new territories.

In particular we will look at random tilings models where the randomness is comes from doubly periodic weights on the underlying bipartite graph and their connection to matrix valued special functions. The project includes the following 6 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 by developing a steepest descent method for their Riemann-Hilbert problem. 3.

Formulate and investigate natural extensions of Schur processes that include doubly periodic weights that have a special integrable structure, such as the two-periodic Aztec diamond. 4. Study the universality of global fluctuations of the height functions. 5.

Prove new Central Limit Theorems fluctuations of linear statistics with for determinantal especially those coming from random tilings. 6. A deeper investigation of the random geometry of the height fluctuations, such as level lines and thick points.