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.
See Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices (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 $n$.
By symmetry you're looking at the probability that the maximal eigenvalue is smaller than some number. Explicit inequalities for such events can be obtained by using the tridiagonal representation for the GOE, see the last 20 slides from Michel Ledoux : https://www.math.univ-toulouse.fr/~ledoux/Leipzig.pdf
