Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

2004/09/28 by Denes Petz, Petz, Denes, Julia Reffy +1
Mathematics · #15A52 #60F10 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:15A52 #msc:60F10

paper · pdf · doi:10.48550/arxiv.math/0409552

arxiv created 2004/10/01 · arxiv updated 2009/12/01

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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