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Kac's random walk on the special orthogonal group mixes in polynomial time


Author: Yunjiang Jiang
Journal: Proc. Amer. Math. Soc. 145 (2017), 4533-4541
MSC (2010): Primary 60J05; Secondary 22C05
DOI: https://doi.org/10.1090/proc/13598
Published electronically: June 22, 2017
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Abstract: We give the first proof of polynomial total variation mixing time bound for the Kac random walk on the special orthogonal groups. The proof relies on the exact spectral gap computation by E. A. Carlen et al. and D. Maslen, and hinges on two novel ingredients: a multi-dimensional generalization of Turan's lemma for polynomials on the unit circle, proved by F. L. Nazarov, and a Morse-theoretic result due to J. Milnor. The techniques are robust in the sense that the step distribution of the walk does not have to be uniformly supported on the circles and that the model can be generalized to higher dimensional particles or other compact Lie groups, provided the corresponding relaxation time is polynomial in the dimension.


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Additional Information

Yunjiang Jiang
Affiliation: Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, California 94305
Address at time of publication: Google Inc., 1600 Amphitheatre Parkway, Mountain View, California 94043
Email: yunjiangster@gmail.com

DOI: https://doi.org/10.1090/proc/13598
Keywords: Kac walk, total variation mixing time, Nazarov's inequality
Received by editor(s): December 15, 2013
Published electronically: June 22, 2017
Additional Notes: The author’s research was partially supported by an NSF graduate fellowship
Dedicated: Dedicated to Professor Theodore Shifrin
Communicated by: Professor Walter Van Assche
Article copyright: © Copyright 2017 American Mathematical Society