The Wigner Theorem states that the statistical distribution of the eigenvalues of a random Hermitian matrix converges to the semi-circular law as the dimension goes to infinity. It is possible to establish this result by using harmonic analysis on the Heisenberg group. In fact this convergence corresponds to the topology of the set of spherical functions associated to the action of the unitary group on the Heisenberg group.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc78-0-9,
author = {Jacques Faraut and Linda Saal},
title = {The Wigner semi-circle law and the Heisenberg group},
journal = {Banach Center Publications},
volume = {75},
year = {2007},
pages = {133-143},
zbl = {1133.15019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc78-0-9}
}
Jacques Faraut; Linda Saal. The Wigner semi-circle law and the Heisenberg group. Banach Center Publications, Tome 75 (2007) pp. 133-143. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc78-0-9/