On pure complex spectrum for truncations of random orthogonal matrices and Kac polynomials

Let $O(2n+\ell)$ be the group of orthogonal matrices of size $\left(2n+\ell\right)\times \left(2n+\ell\right)$ equipped with the probability distribution given by normalized Haar measure. We study the probability \begin{equation*} p_{2n}^{\left(\ell\right)} = \mathbb{P}\left[M_{2n} \, \mbox{has no real eigenvalues}\right], \end{equation*} where $M_{2n}$ is the $2n\times 2n$ left top minor of a $(2n+\ell)\times(2n+\ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $n\times n$ that depends on the truncation parameter $\ell$. For $\ell=1$ the matrix coincides with the Hilbert matrix and we prove \begin{equation*} p_{2n}^{\left(1\right)} \sim n^{-3/8}, \mbox{ when }n \to \infty. \end{equation*} We also discuss connections of the above to the persistence probability for random Kac polynomials.