Acta Mathematica

Volume 227 (2021)

Number 1

The KPZ fixed point

Pages: 115 – 203



Konstantin Matetski (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Jeremy Quastel (Department of Mathematics, University of Toronto, Ontario, Canada)

Daniel Remenik (Departamento de Ingeniería Matemática & Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile)


An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [54], [10]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data.

In the KPZ 1:2:3 scaling limit 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 $\operatorname{Airy}_1$ and $\operatorname{Airy}_2$ processes. The process takes values in real valued functions which look locally like Brownian motion, and is Hölder $\frac{1}{3}-$ 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.

Received 5 December 2018

Received revised 2 April 2020

Accepted 29 September 2020

Published 24 November 2021