Structures and universalities around the Kardar-Parisi-Zhang equation

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It was proposed by Kardar, Parisi and Zhang in the 1980s that a large class of randomly growing interfaces exhibit universal fluctuations described mathematically by a nonlinear stochastic partial differential equation, which is now known as the Kardar-Parisi-Zhang or KPZ equation. Models within this class exhibit three basic mechanisms: growth as a function of the steepness of the interface, a smoothing effect modelled by Laplacian and local randomness modelled by white noise. Examples of physical systems which fall in this class are percolation of liquid in porous media, growth of bacteria colonies, currents in one dimensional traffic or liquid systems, liquid crystals etc.Remarkably the fluctuations of such random interfaces are governed by exponents and distributions that differ from the predictions given by the classical central limit theorem. In dimension one they are, surprisingly, linked to laws emerging from random matrix theory, as this was first exhibited by the work of Baik-Deift-Johansson, followed by a flurry of activity which set the framework of "determinantal processes". New exciting developments have taken place in the more recent years, making the first important steps into universality beyond determinantal models. In dimension two the situation is much less developed as governing exponents and distributions are not known and even the meaning of the two dimensional KPZ is not set in place.The goal of the project is twofold: A. To penetrate deeper into the structure of one dimensional KPZ via setting a robust framework to study fluctuations of non determinantal systems, attacking pending conjectures on multipoint correlations and exploring new grounds into the universality and localisation phenomena. In doing so, novel links between probability, algebraic combinatorics, random matrix theory, integrable systems, number theory (automorphic forms) will be made. B. To make the first steps in dimension two by constructing, via suitable scaling limits of discrete systems, the object(s) that incarnate the two dimensional KPZ equation and extract their properties.