A New Proof of the Abstract Random Tensor Estimate by Deng, Nahmod, and Yue

TL;DR

This paper presents a novel proof of the abstract random tensor estimate, employing non-commutative Khintchine inequality and polynomial tools to simplify and extend the original result by Deng, Nahmod, and Yue.

Contribution

It introduces a new proof technique that removes the square-free restriction and broadens the applicability of the tensor estimate.

Findings

The proof simplifies the original argument.

It extends the estimate to more general Gaussian settings.

The approach leverages advanced polynomial and inequality tools.

Abstract

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 tool in our proof is the direct use of the non-commutative Khintchine inequality with the probabilistic decoupling of the product of Gaussians. Hermite 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.