On the Pseudo-Mixing of Kac's Walk

2026-08-18

AI Summary

研究Kac's walk在SO(n)上的伪混合问题，通过证明前k列在Wasserstein距离下的混合时间上界，解决了Oliveira的猜想，并展示了其在快速Johnson-Lindenstrauss变换中的应用。

Abstract

Motivated by a conjecture of Vaikuntanathan and Zamir, we study the pseudo-mixing of Kac's walk on $\mathrm{SO}(n)$: whether short trajectories are indistinguishable from Haar measure by low-complexity tests. We prove that the first $k$ columns mix in Wasserstein distance in $O(n(k+\log n)\log n)$ steps for fixed accuracy, resolving a conjecture of Oliveira. Combining this with a representation-theoretic variance bound, we show that if $T=ω(nk(k+\log n)\log n)$, then every degree-$k$ polynomial normalized to have unit Haar variance has expectation under the $T$-step law within $o(1)$ of its Haar expectation. As an application, we show that this pseudo-mixing estimate can be used to prove the effectiveness of a fast Johnson--Lindenstrauss transform with the usual target dimension.

Problem

• pseudo-mixing
• Kac's walk
• SO(n)
• Haar measure
• low-complexity tests

Innovation

• Pseudo-mixing
• Kac's Walk
• Wasserstein distance
• Representation-theoretic variance bound
• Johnson-Lindenstrauss transform