AI 中文总结
研究$SO(n)$上坐标平面Kac游走的混合性质，通过正则化逆角校正耦合证明Wasserstein-$2$和全变差$O(n^2)$混合界，并给出二次尺度预截断结果。
AI 中文摘要
我们研究$\mathrm{SO}(n)$上具有独立均匀旋转角的坐标平面Kac游走。对于未归一化的Hilbert-Schmidt黎曼距离，我们证明了当$n$足够大时，$2\binom n2$步一块的Wasserstein-$2$ Lipschitz系数至多为$2n^{-1/200}$。这给出了固定精度的$O(n^2)$ Wasserstein混合界，以及在常数块数内达到逆多项式精度。我们还获得了$O(n^2)$的全变差上界。该耦合将正则化逆角校正的每个分量对其自身角度进行平均，因此其精确流保持联合角度分布。一个确定性递减的正则化参数抵消了主要的标量和矩阵值预解漂移。其稳定性因子是逆正则化的多项式，矩阵估计在两个部分迹上闭合。相同的标量估计界定了不足协方差方向的比例；独立的核和截断分部积分论证给出了全变差转移。我们明确提供了参数估计、条件论证和有限时间奇异质量处理。结果表明在二次尺度上存在全变差预截断，但既未建立截断的存在性也未建立其不存在性。
英文摘要
We study the coordinate-plane Kac walk on $\mathrm{SO}(n)$ with independent uniform rotation angles. For the unnormalized Hilbert-Schmidt Riemannian distance, we prove that the Wasserstein-$2$ Lipschitz coefficient of a block of $2\binom n2$ steps is at most $2n^{-1/200}$ for sufficiently large $n$. This gives a fixed-accuracy $O(n^2)$ Wasserstein mixing bound and inverse-polynomial accuracy in a constant number of blocks. We also obtain an $O(n^2)$ total-variation upper bound. The coupling averages each component of a regularized-inverse angle correction over its own angle, so its exact flow preserves the joint angle law. A deterministic decreasing regularization parameter cancels the main scalar and matrix-valued resolvent drifts. Its stability factor is polynomial in the inverse regularization, and the matrix estimates close on two partial traces. The same scalar estimate bounds the fraction of deficient covariance directions; independent cores and a cutoff integration-by-parts argument give the total-variation transfer. We provide the parameter estimates, conditioning arguments, and finite-time singular-mass treatment explicitly. The results imply total-variation pre-cutoff on the quadratic scale, but do not establish either the presence or the absence of cutoff.
