A new proof of the abstract random tensor estimate by Deng, Nahmod, and Yue

We provide a new proof of the abstract random tensor estimate. This estimate was initially proven by Deng, Nahmod, and Yue (2022) using the moment method. The key new tools are the direct use of the non-commutative Khintchine inequality combined with the probabilistic decoupling of the product of Gaussians. Hermite polynomials and generalized Laguerre-type polynomials allow us to account for pairings in the real and complex-valued Gaussians, respectively, and remove the square-free (tetrahedral) requirement.