A Moment Majorization principle for random matrix ensembles

TL;DR

This paper establishes a moment majorization principle for matrix-valued functions, extending invariance principles to noncommutative settings, with applications in noise stability and anticoncentration.

Contribution

It introduces a noncommutative moment majorization inequality for matrix-valued functions with small influence variables, generalizing classical invariance principles.

Findings

Proves a moment majorization inequality for matrix-valued functions.

Extends invariance principles to noncommutative multilinear polynomials.

Provides applications to noncommutative noise stability and anticoncentration.

Abstract

We prove a moment majorization principle for matrix-valued functions with domain $\{-1,1\}^{m}$, $m\in\mathbb{N}$. The principle is an inequality between higher-order moments of a non-commutative multilinear polynomial with different random matrix ensemble inputs, where each variable has small influence and the variables are instantiated independently. This technical result can be interpreted as a noncommutative generalization of one of the two inequalities of the seminal invariance principle of Mossel, O'Donnell and Oleszkiewicz. Applications to noncommutative noise stability and noncommutative anticoncentration are given.