Department Colloquium: Leonid Petrov

I will survey the general phenomenon of "integrable" probabilistic models, in which the presence of explicit formulas describing their distributions allow for an analysis by essentially algebraic methods.

Then I will discuss in detail an integrable probabilistic model of randomly tiling a hexagon drawn on the regular triangular lattice by lozenges of three types (equivalent formulations: dimer models on the honeycomb lattice, or random 3D stepped surfaces glued out of 1x1x1 boxes). This model has received a significant attention over the past 20 years (first results - the computation of the partition function - date back to P. MacMahon, 100+ years ago). Kenyon, Okounkov, and their co-authors (1998-2007) proved the law of large numbers: when the polygon is fixed and the mesh of the lattice goes to zero, the random 3D surface concentrates around a deterministic limit shape, which is algebraic. I will discuss finer asymptotics: local geometry, behavior of interfaces between phases (which manifests the Kardar-Parisi-Zhang universality), and global fluctuations of random surfaces (described by the Gaussian Free Field), as well as dynamical models associated with random tilings.