An Exercise(?) in Fourier Analysis on the Heisenberg Group
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 263-288.

Soit $H\left(n\right)$ le groupe des matrices supérieures $3×3$ ne contenant que des 1 sur la diagonale et dont les entrées appartiennent à $ℤ/nℤ$, l’anneau des entiers modulo $n$. On montre que la marche aléatoire simple y converge vers la probabilité uniforme en un temps d’ordre ${n}^{2}$. La preuve utilise l’analyse de Fourier et, curieusement, n’est pas immédiate. De nouvelles techniques sont introduites pour borner le spectre, qui sont utiles pour d’autres exemples de marches aléatoires sur des groupes.

Let $H\left(n\right)$ be the group of $3×3$ uni-uppertriangular matrices with entries in $ℤ/nℤ$, 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.

Publié le :
DOI : https://doi.org/10.5802/afst.1533
@article{AFST_2017_6_26_2_263_0,
author = {Bump, Daniel and Diaconis, Persi and Hicks, Angela and Miclo, Laurent and Widom, Harold},
title = {An Exercise(?) in Fourier Analysis on the Heisenberg Group},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {263--288},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 26},
number = {2},
year = {2017},
doi = {10.5802/afst.1533},
language = {en},
url = {http://www.numdam.org/articles/10.5802/afst.1533/}
}
Bump, Daniel; Diaconis, Persi; Hicks, Angela; Miclo, Laurent; Widom, Harold. An Exercise(?) in Fourier Analysis on the Heisenberg Group. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 263-288. doi : 10.5802/afst.1533. http://www.numdam.org/articles/10.5802/afst.1533/

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