On the Pseudo-Mixing of Kac's Walk

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 $T=\omega(nk(k+\log n)\log n)$, 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.

Probability (math.PR); Cryptography and Security (cs.CR); Machine Learning (cs.LG)

arXiv:2608.17374 [math.PR]

(or arXiv:2608.17374v1 [math.PR] for this version)

https://doi.org/10.48550/arXiv.2608.17374 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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