In this paper the authors study the “KPZ fixed point”, supposedly the scaling limit of the solution of the KPZ equation and a whole class of other models supposedly in the same universality class. The KPZ equation has been a major challenge both for physicists and mathematician – prominent names such as Nobel Prize winner Parisi and Fields Medalists Hairer have been associated with these problems. KPZ presented a new kind of behaviour, showing up in many different contexts, but it remained for quite some time not clear how to define it mathematically. The limit object, the KPZ fixed point, is here constructed as the scaling limit of the TASEP (Totally Asymmetric Simple Exclusion Process) which is exactly solvable. The solution method which here turns out to be especially helpful relies on biorthogonal ensembles. Various explicit formulas, as well as properties of the KPZ fixed point, which is a space-time Markov process, are derived. The result appears to be of serious interest.
From the abstract: The formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Brownian motion, for the transition probabilities of the scaling invariant Markov process at the centre of the KPZ universality class. The formula readily reproduces known special self-similar solutions such as the Airy\(_1\) and Airy\(_2\) processes. The process takes values in real-valued functions which look locally like Brownian motion, and is Hölder 1/3 in time. Both the KPZ fixed point and TASEP are shown to be stochastic integrable systems in the sense that the time evolution of their transition probabilities can be linearized through a new Brownian scattering transform and its discrete analogue.

Reviewer: A. C. D. van Enter (Groningen)