Universality in the Kardar–Parisi–Zhang class

forthcoming

This proposal aims to solve central open problems in the study of the Kardar–Parisi–Zhang (KPZ) class as well as develop new tools to study its models. The KPZ class consists of a large set of models, sharing common microscopic dynamical features and aiming at modeling the physical phenomena of random growth interfaces. The two main goals in my proposal are 1) Show that the KPZ-scaling is universal in a large family of non-integrable models in the KPZ class. It is believed that models in the KPZ class should converge, under what is called the KPZ-scaling, to a universal random object. Although widely believed, that the KPZ-scaling is universal is still largely conjectural and was shown mostly for a handful of exactly-solvable (integrable) models using deep connections to representation theory, combinatorics and random matrix theory. My plan is to show that, under the assumption of differentiability and curvature of the shape function, the KPZ-scaling exponents are universal in a large class of non-integrable models called Last Passage Percolation. The principal tools to address this problem will be probabilistic-geometric in nature, using objects such as geodesics and ideas from queuing theory. 2) Construct a new limiting object that will open new paths to study the KPZ equation. The KPZ equation is a central model in the KPZ class for a couple of reasons - firstly, it is believed to be the universal scaling limit in the intermediate disorder regime, and secondly, it is the crossover between two important physical phenomena - the Edwards-Wilkinson class and the KPZ class. I propose to introduce a new scaling limit that will shed light and enable us to study in new ways these two important aspects of the KPZ equation, as well provide new tools to address existing open problems. To achieve this, I plan to take the perspective of scaling limits of random walks in random environment (RWRE), and develop new and complementing tools within that theory.