Two-dimensional edge limits

This project investigates how large random systems organize themselves and why many seemingly unrelated systems exhibit remarkably similar behavior. A central theme is the formation of sharp boundaries separating highly ordered regions from regions with strong random fluctuations. Such phenomena arise in models of crystal growth and ice formation, while the same mathematical structures also arise in methods for extracting signals from complex, high-dimensional data. The project advances the progress of science by discovering new universal laws governing fluctuations near these boundaries and explaining why apparently unrelated models exhibit the same edge behavior. It also promotes national prosperity by translating these mathematical discoveries into statistical software for more reliable analysis of high-dimensional data, which is closely related to machine learning and artificial intelligence. Educational activities include mentoring students through research and simulation projects and developing accessible textbooks and other instructional materials that make this rapidly growing area of mathematics more accessible. The research draws on methods from integrable probability, random matrix theory, algebraic combinatorics, and representation theory to study edge phenomena in two-dimensional random systems. It investigates the shapes and fluctuations of arctic curves in lattice models of statistical mechanics, establishes intrinsic characterizations and universality results for the Airy-beta line ensemble, and studies the ensemble's deformations together with local edge limits in the six-vertex model. A complementary direction explores how these universal edge limits improve statistical inference in high-dimensional signal-plus-noise models. In particular, the project develops uniform formulas for transition distributions and the Airy-Green function, leading to more accurate confidence intervals for signal strengths near critical detection thresholds in spiked covariance and related random matrix models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.