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For a complex $n \times n$ matrix $A$, its numerical range is the set

$$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$

We can further define the smallest absolute value of the numbers in the numerical range as $$r(A) = \inf\ \{ |\lambda| : \lambda \in W(A) \} = \inf_{\|x\|=1} |\langle Ax, x \rangle|.$$

$M=(m_{l,k})_{n\times n}$ is called non-self adjoint Gaussian random matrix if $m_{l,k}$ are i.i.d. standard complex Gaussians $~N(0,1/n)+iN(0,1/n)$.

We are interested in the probability density function of $r(M)$ as $M$ being distributed as non-self adjoint Gaussian random matrix.

What is the probability that $r(M)$ is larger than $1-\epsilon$ for some given constant $\epsilon$?

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  • $\begingroup$ I guess you need to be more specific about what you mean by Gaussian matrix. There are are $n^4$ degrees of freedom in choosing a Gaussian probability measure on the vector space of $n\times n$ complex matrices. $\endgroup$
    – Liviu Nicolaescu
    Commented May 6, 2016 at 10:12
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    $\begingroup$ You probably know that the numerical range is convex, so typically 0 belongs to it. So $r=0$ with probability going to $1$ (and probably with probability $\geg 1-e^{-cn^2} $). If you want more precise information, my guess is that the probability that $r>c$ behaves as $e^{-I (c) n^2}$ for some function I. Look at large deviations for random matrices. $\endgroup$
    – Mikael de la Salle
    Commented May 6, 2016 at 12:40
  • $\begingroup$ Thank you! Can you provide some references? I would like to have the exact dependence of the probability and $\epsilon$. $\endgroup$
    – gondolf
    Commented May 7, 2016 at 1:46

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