Grants and Contributions:

Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)

The one dimensional KPZ universality class contains random growth models, directed random polymer free energies, stochastic Hamilton-Jacobi-Bellman equations, stochastic Burgers' equations, stochastically perturbed reaction-diffusion equations, and interacting particle models. At the physical level, KPZ growth appears in phenomena as wide ranging at forest fire fronts, bacterial colony boundaries, liquid crystals, and coffee rings. The class is characterized by the unusual dynamic scaling exponent z=3/2. A number of breakthroughs about 15 years ago led to a few exact distributions of fluctuations for a few models, with conjectural extrapolation to the whole class. The distributions, surprisingly, turned out to be those recently discovered in random matrix theory, and have now been observed in physical experiments. 6 years ago there was a second group of breakthroughs in which several models with adjustable asymmetry were partially solved leading to exact distributions for various initial conditions for the KPZ equation itself, a non-linear stochastic partial differential equation introduced in the mid 80's as a canonical continuum model in the class. Concurrent breakthroughs on the well-posedness of the KPZ equation itself led to a 2014 Fields medal. There has been intense activity both in the mathematics and physics communities, but this is an unusual area where mathematicians have often been able to take the lead from physicists, on physical problems.

A third KPZ revolution is now beginning, as our group has finally been able to access the invariant Markov process behind all the exact formulas, the KPZ fixed point . In the last few decades, progress in probability and statistical physics has been dominated by such integrable fixed points, such as SLE, the Brownian map, and the sine kernel process, which provide explanations for large fluctuation classes. The KPZ fixed point promises to do the same for the KPZ universality class.

The goal of this proposal is to develop the general exact formulas for the KPZ fixed point, to gain insight into the universality of the fluctuations, to extend the weak universality of the KPZ equation itself, to study the crossover of discrete models from entropy solutions of Burgers' equation at the Euler scale, to these new solutions, and to begin to access problems in higher dimensions.