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## Abstract

For a large

where

*n*×*m*Gaussian matrix, we compute the joint statistics, including large deviation tails, of generalized and total variance - the scaled log-determinant*H*and trace*T*of the corresponding*n*×*n*covariance matrix. Using a Coulomb gas technique, we find that the Laplace transform of their joint distribution*Pn*(*h*,*t*) decays for large*n*,*m*(with*c*=*m*/*n*≥ 1 fixed) as*Ρn*(*s*,*w*) ≈ exp (−*βn*^{2}*J*(*s*,*w*)),where

*β*is the Dyson index of the ensemble and*J*(*s*,*w*) is a*β*-independent large deviation function, whichwe compute exactly for any*c*. The corresponding large deviation functions in real space areworked out and checked with extensive numerical simulations. The results are complemented with a finite*n,m*treatment based on the Laguerre-Selberg integral. The statistics of atypically small log-determinants is shown to be driven by the split-off of the*smallest*eigenvalue, leading to an abrupt change in the large deviation speed.Original language | English |
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Article number | 044306 |

Number of pages | 20 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2016 |

Issue number | 4 |

Early online date | 28 Apr 2016 |

DOIs | |

Publication status | Published - Apr 2016 |

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## Projects

- 1 Finished