Kac's walk on rotation matrices mixes in $n^2 \log n$ steps

2026-04-26

AI Summary

This work addresses the long-standing open problem of determining the total variation mixing time of Kac’s random walk on the rotation group—a challenge exacerbated by the absence of conjugacy invariance—by introducing a novel mixing analysis framework tailored to continuous state spaces and singular transition kernels. The approach employs a two-stage strategy: first, a Wasserstein coupling contracts the chain into a local neighborhood; second, a Malliavin-type analysis expresses the distribution as a pushforward of high-dimensional noise, with matrix martingales and Gaussian approximations on the Lie algebra used to verify the non-degeneracy of the linearized dynamics. This methodology yields the first fine-grained control for non-conjugacy-invariant Markov chains, establishing a total variation mixing time of $O(n^2 \log n)$ steps, thereby matching the conjectured optimal order.

Technology Category

• Machine Learning: Kernel Methods
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Kac's walk on the rotation group, introduced by Hastings in 1970, is an important high-dimensional Markov chain with applications in statistical physics, statistics, cryptography, and computational science. Despite its simple transition rules, determining its total-variation mixing time has remained a challenging problem for decades. A key obstacle is that the walk is not conjugation-invariant, placing it beyond the reach of classical Fourier-analytic techniques that apply to many related random walks on compact groups. We prove that Kac's walk mixes in total variation in \(O(n^2 \log n)\) steps, matching the conjectured mixing time up to constants. The proof is based on a refined two-stage coupling. Building on earlier work, the first stage contracts two copies of the chain to a small neighborhood via a Wasserstein coupling. Our main contribution is a new framework for analyzing the second-stage coupling. It can be viewed as a discrete analogue of Malliavin calculus for Markov chains. We represent the law of the chain as the pushforward of high-dimensional noise and prove quantitative non-degeneracy of the associated linearization using matrix martingale methods. This yields an approximately Gaussian distribution in the Lie algebra with well-conditioned covariance, allowing small group translations to be absorbed at negligible cost in total variation. Our approach provides a general framework for studying mixing in high-dimensional Markov chains in continuous state spaces with singular transition kernels.

Problem

• Kac's walk
• mixing time
• rotation group
• total variation
• Markov chains

Innovation

• Markov chain mixing
• Kac's walk
• coupling
• Malliavin calculus
• matrix martingales