Probabilistic Schubert Calculus: asymptotics

In the recent paper [arXiv:1612.06893] P. Burgisser and A. Lerario introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $delta_{k,n}$ the average number of projective $k$-planes in $mathbb{R}textrm{P}^n$ that intersect $(k+1)(n-k)$ many random, independent and uniformly distributed linear projective subspaces of dimension $n-k-1$. They called $delta_{k,n}$ the expected degree of the real Grassmannian $mathbb{G}(k,n)$ and, in the case $k=1$, they proved that: $$ delta_{1,n}= frac{8}{3pi^{5/2}} cdot left(frac{pi^2}{4}right)^n cdot n^{-1/2} left( 1+mathcal{O}left(n^{-1}right)right) .$$ Here we generalize this result and prove that for every fixed integer $k>0$ and as $nto infty$, we have begin{equation*} delta_{k,n}=a_k cdot left(b_kright)^ncdot n^{-frac{k(k+1)}{4}}left(1+mathcal{O}(n^{-1})right) end{equation*} where $a_k$ and $b_k$ are some (explicit) constants, and $a_k$ involves an interesting integral over the space of polynomials that have all real roots. For instance: $$delta_{2,n}= frac{9sqrt{3}}{2048sqrt{2pi}} cdot 8^n cdot n^{-3/2} left( 1+mathcal{O}left(n^{-1}right)right).$$ Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and we give an explicit formula for $delta_{1,n}$ involving a one dimensional integral of certain combination of Elliptic functions.