I was wondering if there is an explicit estimate on the probability that the lowest eigenvalue of a $n \times n$ GOE matrix is larger than some number $x \in \mathbb{R}$. I am aware of the fact that there is in principle an explicit formula for that, but if $n$ becomes large, this event is really difficult to compute. 


Ideally, there should be also an error bound for that.


Thank you.



Estimate on lowest eigenvalue in GOE
See [Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices](https://arxiv.org/abs/0801.1730) (2008), in particular the large-$n$ result:


$$\text{Prob}(E\_{\rm smallest}\geq x)\rightarrow\exp\left[-n^2\Phi\left(\frac{x+\sqrt{2n}}{\sqrt{n}}\right)\right],\;\;-\sqrt{2n}<x<0,$$
$$\Phi(z)=S(-\sqrt{2})-S(-\sqrt{2}-z),$$
$$S(z)= \frac{1}{216}\left[ 72 z^2 -2z^4 +(30 z + 2z^3)
\sqrt{6 +z^2}+ 27\left( 3 + \ln 1296 - 4
\ln\left(-z + \sqrt{6 +z^2}\right)\right)\right].$$


The probability that all eigenvalues are positive follows from $x\rightarrow 0$, 


$$\text{Prob}(E\_{\rm smallest}\geq 0)\rightarrow3^{-n^2/4}.$$


These are all large-$n$ results: the order $n^2$ exponents have finite-$n$ corrections of order $n$.


