TITLE: Unexpected $\sqrt{3}$
QUESTION [21 upvotes]: A somewhat lengthy calculation, involving integrals, reveals that the probability an $n\times n$ Hermitian matrix, drawn from the Gaussian unitary ensemble, is positive definite, decays as $\left(\sqrt{3}\right)^{-n^2}$ for large $n$. I can’t say what decay constant I had expected, given the context, but it certainly wasn’t $\sqrt{3}$ ! Is there a deeper or more direct explanation for the appearance of this number?

REPLY [26 votes]: this is a limit of a more general result by Majumdar and company, How many eigenvalues of a Gaussian random matrix are positive? (2010), see also their earlier papers from 2006 and 2008.
The coefficient $\sqrt{3}$, or $\frac{1}{2}\log 3$ in the exponent, appears from a saddle point approximation, see Eq. 59 of the 2008 paper, without any particular numerological significance.
Concerning a possible heuristic argument for the number $\sqrt{3}$, here is one published argument that gives $\sqrt{e}$, close but not quite correct.