The KPZ fixed point

TL;DR

This paper derives an explicit Fredholm determinant formula for the multipoint distribution of TASEP's height function, revealing the KPZ fixed point as a universal, stochastic integrable system with Brownian motion-like local behavior.

Contribution

It introduces a new explicit formula for the KPZ fixed point's multipoint distribution and demonstrates its stochastic integrability via a novel Brownian scattering transform.

Findings

Explicit Fredholm determinant formula for TASEP height distribution.

Identification of the KPZ fixed point as a stochastic integrable system.

Reproduction of known Airy processes and local Brownian motion behavior.

Abstract

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 [Sas05; BFPS07]. 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 Airy$_1$ and Airy$_2$ processes. The process takes values in real valued functions which look locally like Brownian motion, and is H\"older $1/3-$ 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.