# CONFLUENTES MATHEMATICI

Principal angles between random subspaces and polynomials in two free projections
Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 3-10.

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 $\left[0,\pi /2\right]$.

Received:
Accepted:
Published online:
DOI: 10.5802/cml.74
Classification: 46L54
Keywords: free probability, principal angles, random subspace
Guillaume Aubrun 1

1 Université de Lyon; CNRS; Université Lyon 1; Institut Camille Jordan UMR5208, 69622 Villeurbanne Cedex, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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
TI  - Principal angles between random subspaces and polynomials in two free projections
JO  - Confluentes Mathematici
PY  - 2021
DA  - 2021///
SP  - 3
EP  - 10
VL  - 13
IS  - 2
PB  - Institut Camille Jordan
UR  - https://cml.centre-mersenne.org/articles/10.5802/cml.74/
UR  - https://doi.org/10.5802/cml.74
DO  - 10.5802/cml.74
LA  - en
ID  - CML_2021__13_2_3_0
ER  - 
%0 Journal Article
%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://doi.org/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] P.-A. Absil; A. Edelman; P. Koev On the largest principal angle between random subspaces, Linear Algebra Appl., Volume 414 (2006) no. 1, pp. 288-294 | DOI | MR

[2] A. Böttcher; I. M. Spitkovsky A gentle guide to the basics of two projections theory, Linear Algebra Appl., Volume 432 (2010) no. 6, pp. 1412-1459 | DOI | MR

[3] Maxime Fevrier; Mitja Mastnak; Alexandru Nica; Kamil Szpojankowski 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] Gene H. Golub; Charles F. Van Loan Matrix computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013, xiv+756 pages | MR

[5] Vladislav Kargin On eigenvalues of the sum of two random projections, Journal of Statistical Physics, Volume 149 (2012) no. 2, pp. 246-258

[6] James A. Mingo; Roland Speicher 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] Alexandru Nica; Roland Speicher Commutators of free random variables, Duke Math. J., Volume 92 (1998) no. 3, pp. 553-592 | DOI | MR

[8] Alexandru Nica; Roland Speicher 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] D. V. Voiculescu; K. J. Dykema; A. Nica 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: