We use the geometric concept of principal angles between subspaces to compute the noncommutative distribution of an expression involving two free projections. For example, this allows to simplify a formula by Fevrier–Mastnak–Nica–Szpojankowski about the free Bernoulli anticommutator. As a byproduct, we observe the remarkable fact that the principal angles between random half-dimensional subspaces are asymptotically distributed according to the uniform measure on .
Accepted:
Published online:
Keywords: free probability, principal angles, random subspace
Guillaume Aubrun 1
CC-BY-NC-ND 4.0
@article{CML_2021__13_2_3_0,
author = {Guillaume Aubrun},
title = {Principal angles between random subspaces and polynomials in two free projections},
journal = {Confluentes Mathematici},
pages = {3--10},
publisher = {Institut Camille Jordan},
volume = {13},
number = {2},
year = {2021},
doi = {10.5802/cml.74},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.74/}
}
TY - JOUR AU - Guillaume Aubrun TI - Principal angles between random subspaces and polynomials in two free projections JO - Confluentes Mathematici PY - 2021 SP - 3 EP - 10 VL - 13 IS - 2 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.74/ DO - 10.5802/cml.74 LA - en ID - CML_2021__13_2_3_0 ER -
%0 Journal Article %A Guillaume Aubrun %T Principal angles between random subspaces and polynomials in two free projections %J Confluentes Mathematici %D 2021 %P 3-10 %V 13 %N 2 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.74/ %R 10.5802/cml.74 %G en %F CML_2021__13_2_3_0
Guillaume Aubrun. Principal angles between random subspaces and polynomials in two free projections. Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 3-10. doi : 10.5802/cml.74. https://cml.centre-mersenne.org/articles/10.5802/cml.74/
[1] On the largest principal angle between random subspaces, Linear Algebra Appl., Volume 414 (2006) no. 1, pp. 288-294 | DOI | MR
[2] A gentle guide to the basics of two projections theory, Linear Algebra Appl., Volume 432 (2010) no. 6, pp. 1412-1459 | DOI | MR
[3] Using Boolean cumulants to study multiplication and anti-commutators of free random variables, Trans. Amer. Math. Soc., Volume 373 (2020) no. 10, pp. 7167-7205 | DOI | MR
[4] Matrix computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013, xiv+756 pages | MR
[5] On eigenvalues of the sum of two random projections, Journal of Statistical Physics, Volume 149 (2012) no. 2, pp. 246-258
[6] Free probability and random matrices, Fields Institute Monographs, 35, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017, xiv+336 pages | DOI | MR
[7] Commutators of free random variables, Duke Math. J., Volume 92 (1998) no. 3, pp. 553-592 | DOI | MR
[8] Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, Cambridge, 2006, xvi+417 pages | DOI | MR
[9] Free random variables, CRM Monograph Series, 1, American Mathematical Society, Providence, RI, 1992, vi+70 pages (A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups) | DOI | MR
Cited by Sources: