### Abstract

We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices A = XX^{∗}, where X is an Nxn matrix with iid standard complex normal entries. Under the scaling n = N+ √4cN®90, c > 0 and N → ∞, we show that the rescaled fluctuations of the smallest eigenvalue, largest eigenvalue and condition number of the matrices A are all given by the Tracy-Widom distribution (β = 2). This scaling is motivated by the study of the solution of the equation Ax = b using the conjugate gradient algorithm, in the case that A and b are random: For such a scaling the fluctuations of the halting time for the algorithm are empirically seen to be universal.

Original language | English (US) |
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Pages (from-to) | 4287-4347 |

Number of pages | 61 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 36 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2016 |

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### Keywords

- Conjugate gradient algorithm
- Laguerre polynomials
- Laguerre unitary ensemble
- Riemann-Hilbert problems
- Wishart ensemble

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*36*(8), 4287-4347. https://doi.org/10.3934/dcds.2016.36.4287