Product of random projections, Jacobi ensembles and universality problems arising from free probability

By Benoît Collins. 2004

132 citations recorded by the source corpus.

Abstract.We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8f2bf1027caaaa95175ce513a3602de2ce2cf1b0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Florent Benaych-Georges
2. Michel Ledoux
3. Raj Rao Nadakuditi
4. Yongcheng Qi
5. Tiefeng Jiang
6. Ioana Dumitriu
7. Alain Rouault
8. Elliot Paquette

In Mathematical Physics

1. Raj Rao Nadakuditi
2. Tiefeng Jiang
3. Florent Benaych-Georges
4. Alain Rouault
5. Ioana Dumitriu
6. Elliot Paquette
7. Yongcheng Qi
8. Michel Ledoux

In Statistics and Probability

1. Tiefeng Jiang
2. Raj Rao Nadakuditi
3. Alain Rouault
4. Florent Benaych-Georges
5. Ioana Dumitriu
6. Elliot Paquette
7. Michel Ledoux
8. Yongcheng Qi

Citations in the corpus

• Deviation Inequalities on Largest Eigenvalues (2007, methodology): “For example, fluctuations of the Jacobi Ensemble constructed over Jacobi polynomials and associated to Beta matrices are addressed in [ Co ].”
• Spectral Radii of Large Non-Hermitian Random Matrices (2014, background): “Collins (2005) proves that A∗A forms a Jacobi ensemble.”
• Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles (2006, background): “Anotheroccurrence ofthe Jacobiensem ble isinteresting (see Doum erc (2005), Collins(2005)).”
• Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles (2009, background): “Proof. By Theorem 5.1 from [ 17 ] or (3.10) from [29], we know (2.4) holds with some unknown”
• The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices (2009, background): “Various extensions of our results to the problem of non-extreme eigenvalues are discussed.”
• GLOBAL FLUCTUATIONS FOR LINEAR STATISTICS OF β-JACOBI ENSEMBLES (2012, background): “Recently, it was shown that in these “classical” cases a different kind of model can be constructed, starting from random projections, rather than random Wishart matrices; or, equivalently, that “chopping off” an appropriate corner of a unitary Haar matrix will yield a matrix whose singular values, ”