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

Let $O(2n+\mathrm{\ell })$ be the group of orthogonal matrices of size $(2n+\mathrm{\ell })\times (2n+\mathrm{\ell })$ 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 $(2n+\mathrm{\ell })\times (2n+\mathrm{\ell })$ 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 $\mathrm{\ell }$. For $\mathrm{\ell }=1$ 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.