Probabilistic Schubert Calculus: Asymptotics

In the recent paper Bürgisser and Lerario (Journal für die reine und angewandte Mathematik (Crelles J), 2016) 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 \({\mathbb {R}}\mathrm {P}^n\) 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:

$$\begin{aligned} \delta _{1,n}= \frac{8}{3\pi ^{5/2}} \cdot \left( \frac{\pi ^2}{4}\right) ^n \cdot n^{-1/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$

Here we generalize this result and prove that for every fixed integer \(k>0\) and as \(n\rightarrow \infty \), we have

$$\begin{aligned} \delta _{k,n}=a_k \cdot \left( b_k\right) ^n\cdot n^{-\frac{k(k+1)}{4}}\left( 1+{\mathcal {O}}(n^{-1})\right) \end{aligned}$$

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:

$$\begin{aligned} \delta _{2,n}= \frac{9\sqrt{3}}{2048\sqrt{2\pi }} \cdot 8^n \cdot n^{-3/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$

Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for \(\delta _{1,n}\) involving a one-dimensional integral of certain combination of Elliptic functions.