Mathematics Colloquium: Random Lozenge Waterfall

Speaker: Leonid Petrov, University of Virginia

Title: Random Lozenge Waterfall

Abstract:  How does the corner of a crystal—like a sugar cube—acquire its rounded shape? This question serves as our entry point into the world of random lozenge tilings, which can be visualized as discrete 3D stepped surfaces. In the first part of this talk, we take a visual journey through the law of large numbers (yielding deterministic limit shapes) and local behavior of lozenge tilings.

In the second part, we explore what happens when we deform the underlying distribution to the q-Racah model. While the classical behavior as q -> 1 is well-understood, keeping q fixed while scaling the system reveals a striking new phenomenon: the "Waterfall." This is a flat limit shape not parallel to any coordinate plane, containing a one-dimensional random stepped interface, in which the usual q -> 1 Gibbs measures collapse into a new universal one-dimensional process. The second part is based on the recent joint work with Alisa Knizel.

Time:  Friday, January 23, 3:30pm – 4:30pm

Place:  Exploratory Hall, room 4106 or Zoom