Probabilistic Schubert Calculus: asymptotics

TL;DR

This paper extends the probabilistic analysis of real Schubert problems, deriving asymptotic formulas for expected degrees of Grassmannians, and reveals their connection to periods and elliptic integrals.

Contribution

It generalizes the asymptotic formula for expected degrees of Grassmannians to all k, introduces explicit constants, and links these to periods and elliptic functions.

Findings

Derived asymptotic formulas for elta_{k,n} as n    for all fixed k.

Explicit constants a_k and b_k involve integrals over polynomials with real roots.

Established that elta_{k,n} are periods in the sense of Kontsevich and Zagier.

Abstract

In the recent paper [arXiv:1612.06893] P. B\"urgisser 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: $$ \delta_{1,n}= \frac{8}{3\pi^{5/2}} \cdot \left(\frac{\pi^2}{4}\right)^n \cdot n^{-1/2} \left( 1+\mathcal{O}\left(n^{-1}\right)\right) .$$ Here we generalize this result and prove that for every fixed integer $k>0$ and as $n\to \infty$, we have \begin{equation*} \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{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.