Assuming $b>0$, $\mathbf{A} \in \mathbb{C}^{m\times n}$ with $m\leq n$ and each element of $\mathbf{A}$ is i.i.d. $\mathcal{CN}(0,1)$ distributed, how to obtain $\mathbb{P} \left[ \det\left( \mathbf{A}\mathbf{A}^H \right) < b \right]$?
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1$\begingroup$ Here's a characterization of the distribution in terms of $\chi^2$: mathoverflow.net/a/268156/4600 $\endgroup$– Bjørn KjosHanssenJan 7 at 8:04

$\begingroup$ It means that the determinant is distributed as the product of independent random variables with a chisquared distribution, but I don't know that how to obtain the CDF of the product of independent random variables with a chisquared distribution. $\endgroup$– sunflower0515Jan 7 at 8:16

$\begingroup$ the linked answer gives the asymptotics of the distribution for large $m$ and $n$; if $m$ and $n$ are small you can compute the CDF directly from the $\chi^2$ distributions; if $n,m$ are neither small nor large then there is no practical way to obtain a closedform expression for the CDF. $\endgroup$– Carlo BeenakkerJan 7 at 13:32
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