On the equal time two-point distribution of the one-dimensional KPZ equation by replica

TL;DR

This paper proves that the two-point distribution of the 1D KPZ equation in the long-time limit matches the Airy$_2$ process, confirming a conjecture and connecting two different mathematical representations.

Contribution

The paper demonstrates the equivalence between Dotsenko's Fredholm determinant formula and the Airy$_2$ process for the KPZ two-point distribution.

Findings

Confirmed the equivalence of two Fredholm determinant formulas

Connected KPZ two-point distribution to the Airy$_2$ process

Validated Dotsenko's formula in the long-time limit

Abstract

In a recent contribution, Dotsenko establishes a Fredholm determinant formula for the two-point distribution of the KPZ equation in the long time limit and starting from narrow wedge initial conditions. We establish that his expression is identical to the Fredholm determinant resulting from the Airy$_2$ process.