Probabilistic Schubert Calculus: Asymptotics

Abstract

AbstractIn 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 $$\delta _{k,n}$$ δ k , n the average number of projective k-planes in $${\mathbb {R}}\mathrm {P}^n$$ R P n that intersect $$(k+1)(n-k)$$ ( k + 1 ) ( n - k ) many random, independent and uniformly distributed linear projective subspaces of dimension $$n-k-1$$ n - k - 1 . They called $$\delta _{k,n}$$ δ k , n the expected degree of the real Grassmannian $${\mathbb {G}}(k,n)$$ G ( k , n ) and, in the case $$k=1$$ k = 1 , they proved that: $$\begin{aligned} \delta _{1,n}= \frac{8}{3\pi ^{5/2}} \cdot \left( \frac{\pi ^2}{4}\right) ^n \cdot n^{-1/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$ δ 1 , n = 8 3 π 5 / 2 · π 2 4 n · n - 1 / 2 1 + O n - 1 . Here we generalize this result and prove that for every fixed integer $$k>0$$ k > 0 and as $$n\rightarrow \infty $$ n → ∞ , we have $$\begin{aligned} \delta _{k,n}=a_k \cdot \left( b_k\right) ^n\cdot n^{-\frac{k(k+1)}{4}}\left( 1+{\mathcal {O}}(n^{-1})\right) \end{aligned}$$ δ k , n = a k · b k n · n - k ( k + 1 ) 4 1 + O ( n - 1 ) where $$a_k$$ a k and $$b_k$$ b k are some (explicit) constants, and $$a_k$$ a k involves an interesting integral over the space of polynomials that have all real roots. For instance: $$\begin{aligned} \delta _{2,n}= \frac{9\sqrt{3}}{2048\sqrt{2\pi }} \cdot 8^n \cdot n^{-3/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$ δ 2 , n = 9 3 2048 2 π · 8 n · n - 3 / 2 1 + O n - 1 . Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for $$\delta _{1,n}$$ δ 1 , n involving a one-dimensional integral of certain combination of Elliptic functions.