Project description
Enhancing our understanding of random tilings of planar domains
Much as tiles are placed on a wall or floor, tilings of planar regions – as the name implies, covering a 2D region with certain shapes with no gaps or overlaps – is an active area of research in mathematics, physics and computer science. Random tilings in which the pattern is a random combination of all possible tiling combinations have important applications in theoretical physics and statistical mechanics. The EU-funded PiRaT project is studying the exotic patterns in the random tiling of planar domains, seeking to expand and enhance random tiling models and shed light on several related conjectures that remain elusive.
Objective
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.
Fields of science
Programme(s)
Funding Scheme
ERC-COG - Consolidator GrantHost institution
100 44 Stockholm
Sweden