Spectral Radii of Large Non-Hermitian Random Matrices

Research output: Contribution to journalArticle

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Abstract

By using the independence structure of points following a determinantal point process, we study the radii of the spherical ensemble, the truncation of the circular unitary ensemble and the product ensemble with parameters n and k. The limiting distributions of the three radii are obtained. They are not the Tracy–Widom distribution. In particular, for the product ensemble, we show that the limiting distribution has a transition phenomenon: When k/ n→ 0 , k/ n→ α∈ (0 , ∞) and k/ n→ ∞, the liming distribution is the Gumbel distribution, a new distribution μ and the logarithmic normal distribution, respectively. The cumulative distribution function (cdf) of μ is the infinite product of some normal distribution functions. Another new distribution ν is also obtained for the spherical ensemble such that the cdf of ν is the infinite product of the cdfs of some Poisson-distributed random variables.

Original languageEnglish (US)
Pages (from-to)326-364
Number of pages39
JournalJournal of Theoretical Probability
Volume30
Issue number1
DOIs
StatePublished - Mar 1 2017

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Non-Hermitian Matrix
Spectral Radius
Random Matrices
Ensemble
Infinite product
Cumulative distribution function
Limiting Distribution
Gaussian distribution
Radius
Gumbel Distribution
Normal Function
Point Process
Truncation
Siméon Denis Poisson
Logarithmic
Distribution Function
Random variable
Distribution function

Keywords

  • Determinantal point process
  • Eigenvalue
  • Extreme value
  • Independence
  • Non-Hermitian random matrix
  • Spectral radius

Cite this

Spectral Radii of Large Non-Hermitian Random Matrices. / Jiang, Tiefeng; Qi, Yongcheng.

In: Journal of Theoretical Probability, Vol. 30, No. 1, 01.03.2017, p. 326-364.

Research output: Contribution to journalArticle

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