Summary

Kardar, Parisi and Zhang (KPZ) proposed in 1986 [1] an equation describing the scale invariance of various phenomena. This field has witnessed in recent years a second youth with decisive theoretical progress [2], such as the identification of universal distributions, and with the demonstration of the emergence of KPZ physics in new experimental systems, such as turbulent liquid crystals [4], and even quantum systems (cold atoms [5], quantum magnets [6], excitons-polaritons [7]). Our goal is to provide a broad and accessible overview of these advances.

Context and motivations

The Kardar–Parisi–Zhang equation [1] was originally solely written to account for a specific phenomenon: the stochastic evolution of certain out-of-equilibrium interfaces and their kinetic roughening. Its properties turned out to apply to much more general problems:  the KPZ equation is now regarded as the starting point for the study of a broad class of universal phenomena, whose scaling laws are described by exponents common to many phenomena, both in classical physics [4] and in quantum physics [5,6,7]. An understanding of this comes in part from the analytical [2,3,9] or numerical [8] solution of models in mathematics and theoretical physics.

Since the early 2000s, it has been realized that this equation belongs to a very broad class of universality of phenomena [3,4] exhibiting scale invariance not only on average, but also by its distributions. At the theoretical level, certain tools from mathematics [2,9] (random matrices, Tracy–Widom distributions) and mathematical physics (integrability) have enabled spectacular advances: it has been observed that very different models at the microscopic scale present the same distributions at the steady state approach. At the experimental level, a set of remarkable experiments has made it possible to highlight the reality of these universal distributions – and this in phenomena ranging from instabilities in liquid crystals [4] to exciton-polariton condensates [7], or transport in magnetic systems [6] in low dimension. A variety of numerical approaches have also emerged, ranging from importance sampling [8] to the modelling of experimentally realistic systems [4,7].

In the last five years, the unexpected emergence of universal KPZ properties in open quantum systems has further renewed interest in this equation, while opening up new theoretical and experimental challenges. For example, in exciton-polariton condensates, the field which follows a KPZ dynamics is a phase field, compact by nature unlike the height field which describes the interface, and the KPZ dynamics in this case intertwines with that of topological defects, leading to a great wealth of new phenomena related to KPZ. Their understanding requires a deep knowledge of both the fundamental aspects of the KPZ equation and also of the theory of open quantum systems. Many advances in this new field rely on numerical simulations of the stochastic Gross–Pitaevskii equation, or a generalized version of it, to account for drive and dissipation in the case of exciton-polariton condensates.

This motivates us to organize a school providing a global overview of these novel results, accounting for their remarkable variety.