Hi,

I have a question about the circular law. Consider $A_n=[x_{ij}]$ a sequence of random matrices where $x_{ij}$ are iid with mean $0$ and variance $1$. Consider $\lambda_{n,1},\dots,\lambda_{n,n}$ the eigenvalues of $\frac{1}{\sqrt{n}} A_n$ and $\mu_n$ the random measure $\frac{1}{n}\sum_{k=1}^n \delta_{\lambda_{n,k}}$ on $\mathbb{C}$. The circular law (see the paper of Tao-Vu-Krishnapur, 2010, annals of proba) states that for each test function $f:\mathbb{C}\rightarrow \\mathbb{C}$ one has almost surely $$ \lim\limits_{n\rightarrow +\infty} \int_{\mathbb{C}} f(z) d\mu_n = \frac{1}{\pi} \int_{p^2+q^2\leq 1} f(p+iq) dpdq $$

By test function, we mean continuous and compactly supported. Is the same conclusion also true if $f$ is the caracteristic function of a closed subset of $\mathbb{C}$ ? Here is a simple example, consider a closed subset $F\subset \mathbb{C}$ with zero Lebesgue measure, is it true that $$\lim\limits_{n\rightarrow +\infty} \mathbb{P}( \forall k \in [1,n], \lambda_{n,k}\in F ) =0 $$

Thanks,

Butthis only gives that $\mathbb E\mu_{n}(F)\to 0$, which is a weaker conclusion than you asked for. – Anthony Quas Mar 11 '13 at 16:24