The main results of the project belong to the area of 2d statistical mechanics and describe macroscopic fluctuations in very large stochastic systems. Examples of the systems are the six-vertex model (which models crystal lattices with hydrogen bonds, e.g. the ice H2O or the potassium dihydrogen phosphate KH2PO4), random stepped surfaces, random matrices(used, in particular, both in high-energy physics and in high-dimensional statistics).

The attached images show an example. One can see there a stepped surface arising from uniformly random lozenge tiling of a hexagon and the field of (centered) macroscopic fluctuations of its height function.

All these systems exhibit a Law of Large Numbers, which means that on the macroscopic scalewith overwhelming probability they become non-random and are described by smooth limit shapes.The major part of the project concentrated on studying the fluctuations around such limit shapes.Our study revealed the universality phenomena here, i.e. that the high-level nature of the fluctuations does not depend on details in the system specification. Along these lines we found Gaussian fluctuations with covariance described by a universal object - 2d Gaussian Free Field in wide classes of models: eigenvalues of corners of random matrices, random lozenge and domino tilings, discrete log-gases. Unexpectedly, we also found that in a specific instance of the six-vertex model called the stochastic six-vertex model, the fluctuations are instead governed by the Tracy-Widom distribution -another universal object, which often arises in the study of the interacting particle systems.This change of behavior from one universality class to another is an important and intriguingquestion for the future research.

On the methods side, the results are based on many new exact formulas that we discovered in the analysis of these models. For tilings and random matrix objects, we introduced a new method relying on the applications of differential operators to partition functions withinhomogneities. This can be treated as the development of the Fourier transform for the unitary groups as their rank goes to infinity. For the related framework of discrete log-gases we introduced a new powerful tool of discrete Schwinger-Dyson equations, originating in the work of Nekrasov in theoretical physics. Finally, the developments for the six-vertex model combined the classical Bethe ansatz method with summation identities recently discovered in the study of interacting particle systems. The ongoing work combines all these methods together in order to reach the asymptotic of even more extensive classes of models.

Besides purely scientific developments, the PI organized a conference "Non-equilibrium dynamics of stochastic and quantum integrablesystems" (Kavli Institute, Santa Barbara, USA, Spring 2016), invited session "Integrable Probability" at Stochastic Processes and Applications-2017 conference (Moscow,Russia, Summer 2017), and numerous research seminars to stimulate collaborations, promote and disseminate the results in the wide research area of the project.

Last Modified: 09/02/2017
Modified by: Vadim Gorin