Harmonicity and invariance on slices of the Boolean cube

TL;DR

This paper proves an invariance principle for low-degree functions on the Boolean slice, showing their distributions are similar under slice and cube measures, using novel martingale decompositions.

Contribution

It extends invariance principles to general low-degree functions on the slice without influence constraints, introducing new martingale-based proof techniques.

Findings

Functions with degree o(√n) have similar distributions on slice and cube.

New martingale decomposition for non-reversible, non-stationary Markov chains.

Demonstrates indistinguishability of slice and cube based on few coordinates.

Abstract

In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree functions. Here we provide an alternative proof for general low-degree functions, with no constraints on the influences. We show that any real-valued function on the slice, whose degree when written as a harmonic multi-linear polynomial is $o(\sqrt{n})$, has approximately the same distribution under the slice and cube measure. Our proof is based on a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, ours decomposition is applied in a non-reversible non-stationary setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof. Finally, we provide independent simple proofs for the facts that 1) one cannot distinguish between the slice and the cube based on functions of $o(n)$ coordinates and 2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is $o(\sqrt{n})$.