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Daniel Bump; Persi Diaconis; Angela Hicks; Laurent Miclo; Harold Widom
An Exercise(?) in Fourier Analysis on the Heisenberg Group
Annales de la faculté des sciences de Toulouse Sér. 6, 26 no. 2 (2017), p. 263-288, doi: 10.5802/afst.1533
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Résumé - Abstract

Let $H(n)$ be the group of $3\times 3$ uni-uppertriangular matrices with entries in ${\mathbb{Z}}/{n\mathbb{Z}}$, the integers mod $n$. We show that the simple random walk converges to the uniform distribution in order $n^2$ steps. The argument uses Fourier analysis and is surprisingly challenging. It introduces novel techniques for bounding the spectrum which are useful for a variety of walks on a variety of groups.

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