Probabilistic Schubert Calculus: asymptotics

Abstract

In the recent paper [arXiv:1612.06893] P. Bürgisser 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 $\mathbb{R}{\text{P}}^n$ 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: $${\delta }_{1,n}=\frac{8}{3{\pi }^{5/2}}\cdot {\left(\frac{{\pi }^2}{4}\right)}^n\cdot n^{-1/2}\left(1+\mathcal{O}\left(n^{-1}\right)\right).$$ Here we generalize this result and prove that for every fixed integer $k>0$ and as $n\to \infty$, we have \begin{equation*} δ_{k,n}=a_k \cdot \left(b_k\right)^n\cdot 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: $${\delta }_{2,n}=\frac{9\sqrt{3}}{2048\sqrt{2\pi }}\cdot 8^n\cdot n^{-3/2}\left(1+\mathcal{O}\left(n^{-1}\right)\right).$$ 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.