Resolvent characteristics and quadratic mixing of Kac's walk on SO(n)
Abstract
We study the coordinate-plane Kac walk on 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 2binom n2 steps is at most 2n−1/200 for sufficiently large n. This gives a fixed-accuracy O(n2) Wasserstein mixing bound and inverse-polynomial accuracy in a constant number of blocks. We also obtain an O(n2) 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.
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BibTeX
@misc{jiang2026resolvent,
title = {{Resolvent characteristics and quadratic mixing of Kac's walk on $SO(n)$}},
author = {Yunjiang Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08018},
eprint = {2609.08018},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08018},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Resolvent characteristics and quadratic mixing of Kac's walk on SO(n)
AU - Yunjiang Jiang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.08018
UR - https://arxiv.org/abs/2609.08018
AB - We study the coordinate-plane Kac walk on 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 2binom 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.
ER -