Moments Match between the KPZ Equation and the Airy Point Process

TL;DR

This paper proves a direct combinatorial correspondence between the moments of the KPZ equation's solution and the Airy point process, confirming their distributional equivalence.

Contribution

It offers a straightforward combinatorial proof of moment identities linking the KPZ equation and the Airy point process, simplifying previous complex derivations.

Findings

Moment identities match between KPZ solution and Airy process

Direct combinatorial proof established for distributional equivalence

Simplifies understanding of KPZ-Airy connection

Abstract

The results of Amir-Corwin-Quastel, Calabrese-Le Doussal-Rosso, Dotsenko, and Sasamoto-Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple direct proof of those via a combinatorial match of their multivariate integral representations.