Chaos for rescaled measures on Kac's sphere

TL;DR

This paper investigates the chaotic behavior of rescaled probability measures on Kac's sphere, demonstrating convergence in multiple senses and providing explicit polynomial rates, while improving previous entropic chaos results.

Contribution

It introduces a new approach to constructing chaotic sequences on Kac's sphere and extends existing entropic chaos results by relaxing moment conditions.

Findings

Sequences are chaotic in Wasserstein, $L^1$, entropic, and Fisher information senses.

Explicit polynomial rates of convergence are established.

Improves previous results by reducing moment requirements from 6 to 4+epsilon.

Abstract

In this article we study a relatively novel way of constructing chaotic sequences of probability measures supported on Kac's sphere, which are obtained as the law of a vector of $N$ i.i.d. variables after it is rescaled to have unit average energy. We show that, as $N$ increases, this sequence is chaotic in the sense of Kac, with respect to the Wasserstein distance, in $L^1$, in the entropic sense, and in the Fisher information sense. For many of these results, we provide explicit rates of polynomial order in $N$. In the process, we improve a quantitative entropic chaos result of Haurey and Mischler by relaxing the finite moment requirement on the densities from order $6$ to $4+\epsilon$.