Scaling limits of random plane partitions and six-vertex models

Abstract

We present a collection of results about the scaling limits of several models from integrable probability. Our first result concerns the asymptotic behavior of the bottom slice of a Hall-Littlewood random plane partition. We show the latter concentrates around a limit shape and in two different scaling regimes identify the fluctuations around this shape with the GUE Tracy- Widom distribution and the narrow wedge initial data solution to the Kardar-Parisi-Zhang (KPZ) equation. The second result concerns the limiting behavior of a class of six-vertex models in the quadrant, and we obtain the GUE-corners process as a scaling limit for this class near the boundary. Our final result, joint with Ivan Corwin, demonstrates the (long predicted) transversal 2/3 critical exponent for the height functions of the stochastic sixvertex model and asymmetric simple exclusion process (ASEP). The algebraic parts of our arguments involve the construction and use of degenerations and modifications of the Macdonald difference operators to obtain rich families of observables for the models we consider. These formulas are in terms of multiple contour integrals and provide a direct access to quantities of interest. The analytic parts of our arguments include the detailed asymptotic analysis of Fredholm determinants and contour integrals through steepest descent methods. An important aspect of our approach, is the combination of exact formulas with more probabilistic arguments, based on various Gibbs properties enjoyed by the models we study.