Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Determinantal point processes in the plane from products of random matrices

Kartick Adhikari, Nanda Kishore Reddy, Tulasi Ram Reddy, and Koushik Saha

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Abstract

We show that the density of eigenvalues for three classes of random matrix ensembles is determinantal. First we derive the density of eigenvalues of product of $k$ independent $n\times n$ matrices with i.i.d. complex Gaussian entries with a few of matrices being inverted. In second example we calculate the same for (compatible) product of rectangular matrices with i.i.d. Gaussian entries and in last example we calculate for product of independent truncated unitary random matrices. We derive exact expressions for limiting expected empirical spectral distributions of above mentioned ensembles.

Résumé

Nous montrons que la densité des valeurs propres pour trois classes d’ensembles de matrices aléatoires a une forme déterminantale. D’abord nous dérivons la densité des valeurs propres de produits de $k$ matrices $n\times n$ indépendantes avec entrées i.i.d. gaussiennes avec certaines matrices inversées. Dans le deuxième exemple, nous calculons la même densité pour des produits compatibles de matrices rectangulaires avec entrées i.i.d. gaussiennes et dans le dernier exemple pour des produits de matrices unitaires tronquées aléatoires et indépendantes. Nous dérivons des expressions exactes pour les limites des distributions spectrales de ces exemples.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 16-46.

Dates
Received: 21 October 2013
Revised: 22 April 2014
Accepted: 30 June 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089258

Digital Object Identifier
doi:10.1214/14-AIHP632

Mathematical Reviews number (MathSciNet)
MR3449292

Zentralblatt MATH identifier
1192.60067

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60G55: Point processes 15A18: Eigenvalues, singular values, and eigenvectors 15B99: None of the above, but in this section

Keywords
Determinantal point process Eigenvalues Empirical spectral distribution Limiting spectral distribution Haar measure QR decomposition Random matrix RQ decomposition Generalized Schur decomposition Unitary matrix Wedge product

Citation

Adhikari, Kartick; Reddy, Nanda Kishore; Reddy, Tulasi Ram; Saha, Koushik. Determinantal point processes in the plane from products of random matrices. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 16--46. doi:10.1214/14-AIHP632. https://projecteuclid.org/euclid.aihp/1452089258


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