# Fluctuations at the edges of the spectrum of the full rank deformed GUE

Abstract : We consider a full rank deformation of the GUE $W_N+A_N$ where $A_N$ is a full rank Hermitian matrix of size $N$ and $W_N$ is a GUE. The empirical eigenvalue distribution $\mu_{A_N}$ of $A_N$ converges to a probability distribution $\nu$. We identify all the possible limiting eigenvalue statistics at the edges of the spectrum, including outliers, edges and merging points of connected components of the limiting spectrum. The results are stated in terms of a deterministic equivalent of the empirical eigenvalue distribution of $W_N+A_N$, namely the free convolution of the semi-circle distribution and the empirical eigenvalues distribution of $A_N$.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01011501
Contributor : Mireille Capitaine <>
Submitted on : Tuesday, June 24, 2014 - 9:41:03 AM
Last modification on : Friday, March 27, 2020 - 3:24:52 AM

### Citation

Mireille Capitaine, S. Péché. Fluctuations at the edges of the spectrum of the full rank deformed GUE. Probability Theory and Related Fields, Springer Verlag, 2016, 165 (1), pp.117-161. ⟨10.1007/s00440-015-0628-6⟩. ⟨hal-01011501⟩

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