Moments Match between the KPZ Equation and the Airy Point Process

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.