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Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

Publication ,  Journal Article
Cook, NA; Guionnet, A; Husson, J
Published in: Annales De L'Institut Henri Poincare (B) Probability and Statistics
November 1, 2022

For a fixed quadratic polynomial p in n non-commuting variables, and n independent N × N complex Ginibre matrices XN1, ⋯, XNn, we establish the convergence of the empirical measure of the eigenvalues of PN = p(XN1, ⋯, XNn) to the Brown measure of p evaluated at n freely independent circular elements c1, ⋯, cn in a non-commutative probability space. As in previous works on non-normal random matrices, a key step is to obtain quantitative control on the pseudospectrum of PN. Via a linearization trick of Haagerup-Thorbjørnsen for lifting non-commutative polynomials to tensors, we obtain this as a consequence of a lower tail estimate for the smallest singular value of patterned block matrices with strongly dependent entries. This reduces to establishing anticoncentration for determinants of random walks in a matrix space of bounded dimension, for which we encounter novel structural obstacles of an algebro-geometric nature.

Duke Scholars

Published In

Annales De L'Institut Henri Poincare (B) Probability and Statistics

DOI

ISSN

0246-0203

Publication Date

November 1, 2022

Volume

58

Issue

4

Start / End Page

2284 / 2320

Related Subject Headings

  • Statistics & Probability
  • 0104 Statistics
 

Citation

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ICMJE
MLA
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Cook, N. A., Guionnet, A., & Husson, J. (2022). Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices. Annales De L’Institut Henri Poincare (B) Probability and Statistics, 58(4), 2284–2320. https://doi.org/10.1214/21-AIHP1225
Cook, N. A., A. Guionnet, and J. Husson. “Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices.” Annales De L’Institut Henri Poincare (B) Probability and Statistics 58, no. 4 (November 1, 2022): 2284–2320. https://doi.org/10.1214/21-AIHP1225.
Cook NA, Guionnet A, Husson J. Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices. Annales De L’Institut Henri Poincare (B) Probability and Statistics. 2022 Nov 1;58(4):2284–320.
Cook, N. A., et al. “Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices.” Annales De L’Institut Henri Poincare (B) Probability and Statistics, vol. 58, no. 4, Nov. 2022, pp. 2284–320. Scopus, doi:10.1214/21-AIHP1225.
Cook NA, Guionnet A, Husson J. Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices. Annales De L’Institut Henri Poincare (B) Probability and Statistics. 2022 Nov 1;58(4):2284–2320.
Journal cover image

Published In

Annales De L'Institut Henri Poincare (B) Probability and Statistics

DOI

ISSN

0246-0203

Publication Date

November 1, 2022

Volume

58

Issue

4

Start / End Page

2284 / 2320

Related Subject Headings

  • Statistics & Probability
  • 0104 Statistics