The motivation for the PiRaT starts with remarkable the phenomenon where large systems, composed of many small, interacting elements, display universal patterns. These patterns emerge not from the intricate details of each system but from broader, more general properties. Given the inherent complexity of these systems, which often makes detailed analysis challenging, we rely on theoretical frameworks that both reveal such patterns and offer tools for their study. Random tilings and random matrix theory are two key frameworks. The patterns they uncover are deeply connected to variety of other complex systems, ranging from theoretical examples such as the zeros of the Riemann zeta function and quantum systems such the energy levels of heavy nuclei, to applications to wireless communication. The study of random tilings is a very active and competitive area of mathematical research, that has witnessed dramatic progress in the past two decades. There are many important open problems that need to be answered, many structures that need to be better understood, and, very likely, other interesting phenomena to be discovered.
A tiling of a domain is a covering of that domain by smaller given pieces such that no pieces overlap. There are often many ways of doing this, and one may wonder how a randomly chosen tiling looks like. As it turns out, the random tilings are not as chaotic as one might expect. In the figures attached to the document, one finds to samples of domino tilings of the Aztec diamond with different probability measures. Notice how each tiling naturally divides the domain into disordered regions and frozen zones. To a given tiling there is a natural associated height function, which, in turn, gives rise to a random surface Indeed, for large domains, these random surfaces typically seem to concentrate near a deterministic limit shape. What determines the geometry of the limit shape? Can one characterize the boundary between the ordered and disordered regions? Additionally, what are the local random processes that appear within the disordered regions or near the boundaries, and are these processes universal? Finally, can the global fluctuations of the random surface be described, and if so, are they also universal? These are some examples of general questions that one would like to answer. Especially since these phenomena and laws are expected also to appear in other complicated systems.
The overall aim of the PiRaT project is to solve several open problems, make progress on important conjectures and explore new territories. The emphasis is on random tiling models for with non-uniform measures, both with integrable structures, such as doubly periodic weights, and non-integrable structures, such as random weights.
