Periodic Reporting for period 2 - PiRaT (Patterns in Random Tilings)
Periodo di rendicontazione: 2023-09-01 al 2025-02-28
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.
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.
One of the central objectives of the PiRaT project is to investigate random tilings of planar domains influenced by doubly periodic weights. One motivation for studying these weights arises from the emergence of new phases, termed rough disordered or gaseous regimes, which are unexpected for uniform weights. A defining statistical characteristic of these regimes is the exponential decay of dimer-dimer correlations with increasing distance. At the outset of the project, the exploration of doubly periodic weighting for planar domains was largely uncharted territory, since existing techniques were inadequate. There has been lots of recent activity around this topic, with important contributions from the PiRaT team. For the particular case of domino tilings of the Aztec diamond, the doubly periodic weights are now very well understood. Other types of domains still remain an interesting challenge that we plan to work on.
Another key aim of the project is to analyze the fluctuations of the random surfaces associated with these tilings. A well-known conjecture suggests that the global fluctuations of these surfaces are governed by the Gaussian Free Field (GFF) under Dirichlet boundary conditions. This conjecture is particularly fascinating because the GFF is a fundamental object in conformal field theory. While it has been validated for various classes of models, it remains an intriguing open problem. One of the goals of the PiRaT project is to develop new tools to prove this conjecture, particularly by creating techniques for studying determinants of certain operators that would enable us to analyze height fluctuations. We have made significant progress in this area and aim to advance these techniques further.
Random tilings of planar domains exhibit not only complex behaviors but also remarkable structures that connect them to other mathematical disciplines. A noteworthy example related to the PiRaT project is the connection with (matrix-valued) orthogonal polynomials. These classical objects from analysis have gained prominence over the past two decades due to their relevance in stochastic models such as random matrices and random tilings, especially with the advent of modern analytical tools for their study. Within our research team, we have been examining orthogonal polynomials in various contexts to address important open problems in the field.
Another key aim of the project is to analyze the fluctuations of the random surfaces associated with these tilings. A well-known conjecture suggests that the global fluctuations of these surfaces are governed by the Gaussian Free Field (GFF) under Dirichlet boundary conditions. This conjecture is particularly fascinating because the GFF is a fundamental object in conformal field theory. While it has been validated for various classes of models, it remains an intriguing open problem. One of the goals of the PiRaT project is to develop new tools to prove this conjecture, particularly by creating techniques for studying determinants of certain operators that would enable us to analyze height fluctuations. We have made significant progress in this area and aim to advance these techniques further.
Random tilings of planar domains exhibit not only complex behaviors but also remarkable structures that connect them to other mathematical disciplines. A noteworthy example related to the PiRaT project is the connection with (matrix-valued) orthogonal polynomials. These classical objects from analysis have gained prominence over the past two decades due to their relevance in stochastic models such as random matrices and random tilings, especially with the advent of modern analytical tools for their study. Within our research team, we have been examining orthogonal polynomials in various contexts to address important open problems in the field.
While studying the double periodic weightings for domino tilings of the Aztec diamond we found some interesting structures that led us to believe that we even can handle probability measure that do not have an integrable structures. Note that integrable structures are often convenient as they may provide tools to prove interesting phenomena, but since the phenomena are supposed universal they should not depend on integrability. This makes the non-integrable weights very important. Furthermore, they can also lead to more complex behavior. Currently, we are studying two different cases of non-integrable weights for domino tilings of the Aztec diamond: (1) slowly varying weights and (2) random weights. Especially the latter class is intriguing as there exist intriguing prediction form the physics literature that a new phase, the so-called super-rough phase, will appear. Any progress in this direction would be a major breakthrough in the field.
In the second half of the project, we also plan to investigate height fluctuations and their connections to Gaussian Multiplicative Chaos. These connections have been extensively studied in the random matrix community, especially due to their relationship with the zeros of the Riemann zeta function. However, very little is known in the context of two-dimensional models, such as random tilings. This gap may be attributed to the fact that several crucial estimates from the one-dimensional setting do not readily translate to two dimensions. The principal investigator is hopeful that the operator approach developed with coauthors can illuminate this area. While we have already obtained some partial results, significant challenges remain.
In the second half of the project, we also plan to investigate height fluctuations and their connections to Gaussian Multiplicative Chaos. These connections have been extensively studied in the random matrix community, especially due to their relationship with the zeros of the Riemann zeta function. However, very little is known in the context of two-dimensional models, such as random tilings. This gap may be attributed to the fact that several crucial estimates from the one-dimensional setting do not readily translate to two dimensions. The principal investigator is hopeful that the operator approach developed with coauthors can illuminate this area. While we have already obtained some partial results, significant challenges remain.