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July 2017 Cut-off phenomenon in the uniform plane Kac walk
Bob Hough, Yunjiang Jiang
Ann. Probab. 45(4): 2248-2308 (July 2017). DOI: 10.1214/16-AOP1111


We consider an analogue of the Kac random walk on the special orthogonal group $\mathrm{SO}(N)$, in which at each step a random rotation is performed in a randomly chosen 2-plane of $\mathbb{R}^{N}$. We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large $N$ limit. In the special case where the angle of rotation is deterministic, this confirms a conjecture of Rosenthal [Ann. Probab. 22 (1994) 398–423]. Under mild conditions, we also establish a cut-off for convergence of the walk to stationarity under the $L^{2}$ norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and $L^{2}$ norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis.


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Bob Hough. Yunjiang Jiang. "Cut-off phenomenon in the uniform plane Kac walk." Ann. Probab. 45 (4) 2248 - 2308, July 2017.


Received: 1 February 2013; Revised: 1 August 2015; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 06786081
MathSciNet: MR3693962
Digital Object Identifier: 10.1214/16-AOP1111

Primary: 60J05
Secondary: 20C15 , 43A75 , 60B15

Keywords: character theory , cut-off phenomenon , Random walk on a group , saddle point analysis

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 4 • July 2017
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