A Few Ways to Destroy Entropic Chaoticity on Kac's Sphere

TL;DR

This paper explores methods to generate chaotic families on Kac's Sphere that lack entropic chaoticity, including convex combinations with non-entropic families and rapidly varying families affecting support and entropy.

Contribution

It introduces new constructions of chaotic families that are not entropically chaotic, highlighting mechanisms for loss of entropic properties.

Findings

Convex combinations with non-entropic families can destroy entropic chaoticity.

Explicitly computable families with rapid variation cause support loss or high entropic tails.

Demonstrates how certain constructions lead to non-entropic chaotic families.

Abstract

In this work we discuss a few ways to create chaotic families that are not entropically chaotic on Kac's Sphere. We present two types of examples: limiting convex combination of an entropically chaotic family with a particularly 'bad' non-entropic family, and two explicitly computable families that vary rapidly with $N$, causing loss of support on the sphere or high entropic tails.