Abstract
Let U m be an m×m Haar unitary matrix and U[ m,n ] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U[ m,n ] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U[ m,n ] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.
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Petz, D., Réffy, J. Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices. Probab. Theory Relat. Fields 133, 175–189 (2005). https://doi.org/10.1007/s00440-004-0420-5
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DOI: https://doi.org/10.1007/s00440-004-0420-5
Keywords
- Random matrices
- Joint eigenvalue distribution
- Haar unitary
- Truncated Haar unitary
- Large deviation
- Rate function
- Free probability
- Random matrix model