In the past twenty years, there have been huge developments in the study of the Kardar-Parisi-Zhang (KPZ) universality class, which is a broad class of physical and probabilistic models including one-dimensional interface growth processes, interacting particle systems and polymers in random environments, etc. It is broadly believed and partially proved, that all the models share the universal scaling exponents and have the same asymptotic behaviors. The height functions of models in the KPZ universality class are expected to converge to a limiting space-time fluctuation field, which is called the KPZ fixed point. Moreover, there is a random “directed metric” on the space-time plane that is expected to govern all the models in the KPZ universality class. This “directed metric” is called the directed landscape. Both the KPZ fixed point and the directed landscape are central objects in the study of the KPZ universality class, while they were only characterized/constructed very recently [MQR21, DOV18].

In this talk, we will discuss some exact formulas of distributions in these two random fields. These exact formulas are in terms of an infinite sum of multiple contour integrals, which are analogous to the Fredholm determinant expansions. We will show some surprising probabilistic properties of the KPZ fixed point and the directed landscape using the exact formulas. Some of the results are based on joint work with Yizao Wang and Ray Zhang.

A recording of the talk can be found at https://youtu.be/I4zWZXU2ZTY