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A note on the circular law for noncentral random matrice  Djalil Chafai
;  Date: 
1 Sep 2007  Abstract:  Let $(X_{i,j})_{1leq i,j<infty}$ be an infinite array of i.i.d. complex
random variables, with mean $m=0$, variance $si^2=1$, and say with finite
fourth moment. The famous circular law theorem states that the empirical
spectral distribution $frac{1}{n}(de_{la_1(X)}+...+de_{la_n(X)})$ of
$X=(n^{1/2}X_{i,j})_{1leq i,jleq n}$ converges almost surely, as
$n oinfty$, to the uniform law over the unit disc ${zindC;ABS{z}leq
1}$. For now, most efforts where focused on the improvement of moments
hypotheses for the centered case $m=0$. Regarding the noncentral case
$m
eq0$, Silverstein has already observed that almost surely, the eigenvalue
of $X$ of largest module goes to $+infty$ as $n oinfty$, while the rest of
the spectrum remains bounded. We show in this note that the circular law
theorem remains valid when $m
eq0$, by using logarithmic potentials and bounds
on extremal singular values.  Source:  arXiv, 0709.0036  Services:  Forum  Review  PDF  Favorites 


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