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.

翻译：Kac在旋转群上的游走由Hastings于1970年提出，是统计物理、统计学、密码学和计算科学中重要的一种高维马尔可夫链。尽管其转移规则简单，但确定其全变差混合时间数十年来一直是一个具有挑战性的问题。一个关键障碍是，该游走不具有共轭不变性，因此无法应用经典傅里叶分析技术——这类技术对许多紧群上的相关随机游走是有效的。我们证明Kac游走在\(O(n^2 \log n)\)步内达到全变差混合，与猜想混合时间仅相差常数因子。证明基于一种精细的两阶段耦合方法。第一阶段基于先前工作，通过Wasserstein耦合将两条链的副本收缩到一个小邻域内。我们的主要贡献是提出一个分析第二阶段耦合的新框架，可视为马尔可夫链的Malliavin微分离散类比。我们将链的分布表示为高维噪声的推进前向分布，并利用矩阵鞅方法证明相关线性化的定量非退化性。这得到李代数中具有良态协方差的近似高斯分布，从而允许以可忽略的全变差代价吸收小群平移。我们的方法为研究具有奇异转移核的高维连续状态空间马尔可夫链的混合性提供了通用框架。