A note on the circular law for non-central random matrices
Résumé
Let $(X_{i,j})_{1\leq i,j<\infty}$ 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 $\frac{1}{n}(\de_{\la_1(\bX)}+\cdots+\de_{\la_n(\bX)})$ of $\bX=(n^{-1/2}X_{i,j})_{1\leq i,j\leq n}$ converges almost surely, as $n\to\infty$, to the uniform law over the unit disc $\{z\in\dC;\ABS{z}\leq 1\}$. For now, most efforts where focused on the improvement of moments hypotheses for the centered case $m=0$. Regarding the non-central case $m\neq0$, Silverstein has already observed that almost surely, the eigenvalue of $\bX$ of largest module goes to $+\infty$ as $n\to\infty$, while the rest of the spectrum remains bounded. We show in this note that the circular law theorem remains valid when $m\neq0$, by using logarithmic potentials and bounds on extremal singular values.
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