Open Access
September 2021 The KPZ fixed point
Konstantin Matetski, Jeremy Quastel, Daniel Remenik
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Acta Math. 227(1): 115-203 (September 2021). DOI: 10.4310/ACTA.2021.v227.n1.a3


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.


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Konstantin Matetski. Jeremy Quastel. Daniel Remenik. "The KPZ fixed point." Acta Math. 227 (1) 115 - 203, September 2021.


Received: 5 December 2018; Accepted: 29 September 2020; Published: September 2021
First available in Project Euclid: 1 March 2023

Digital Object Identifier: 10.4310/ACTA.2021.v227.n1.a3

Vol.227 • No. 1 • September 2021
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