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02 November 2024
 
  » arxiv » 0709.0036

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A note on the circular law for non-central random matrice
Djalil Chafai ;
Date 1 Sep 2007
AbstractLet $(X_{i,j})_{1leq i,j<infty}$ 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 non-central 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
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