Probabilistic Schubert Calculus

TL;DR

This paper introduces the concept of expected degree in real Grassmannians, linking it to complex degrees and convex geometry, and explores its implications for random enumerative geometry of flats.

Contribution

It defines and analyzes the expected degree of real Grassmannians, providing asymptotic formulas and connecting it to complex degrees and convex bodies, advancing understanding of random intersection problems.

Findings

Expected degree expressed via volume of invariant convex body.

Asymptotic relation between real and complex Grassmannian degrees.

Expected degree governs random enumerative geometry of flats.

Abstract

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree $\textrm{edeg} G(k,n)$ of the real Grassmannian $G(k,n)$ as the average number of real $k$-planes meeting nontrivially $k(n-k)$ random subspaces of $\mathbb{R}^n$, all of dimension $n-k$, where these subspaces are sampled uniformly and independently from $G(n-k,n)$. We express $\textrm{edeg} G(k,n)$ in terms of the volume of an invariant convex body in the tangent space to the Grassmanian, and prove that for fixed $k\ge 2$ and $n\to\infty$, $$ \textrm{edeg} G(k,n) = \textrm{deg} G_\mathbb{C}(k,n)^{\frac{1}{2} \epsilon_k + o(1)}, $$ where $\textrm{deg} G_\mathbb{C}(k,n)$ denotes the degree of the corresponding complex Grassmannian and $\epsilon_k$ is monotonically decreasing with $\lim_{k\to\infty} \epsilon_k = 1$. In the case of the Grassmannian of lines, we prove the finer asymptotic \begin{equation*} \textrm{edeg} G(2,n+1) = \frac{8}{3\pi^{5/2}\sqrt{n}}\, \left(\frac{\pi^2}{4} \right)^n \left(1+\mathcal{O}(n^{-1})\right). \end{equation*} The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set $X\subseteq\mathbb{R}\textrm{P}^{n-1}$ of dimension $n-k-1$ its Chow hypersurface $Z(X)\subseteq G(k,n)$, consisting of the $k$-planes $A$ in $\mathbb{R}^n$ whose projectivization intersects $X$. Denoting $N:=k(n-k)$, we show that $$ \mathbb{E}\#\left(g_1Z(X_1)\cap\cdots\cap g_N Z(X_N)\right) = \textrm{edeg} G(k,n) \cdot \prod_{i=1}^{N} \frac{|X_i|}{|\mathbb{R}\textrm{P}^{m}|}, $$ where each $X_i$ is of dimension $m=n-k-1$, the expectation is taken with respect to independent uniformly distributed $g_1,\ldots,g_m\in O(n)$ and $|X_i|$ denotes the $m$-dimensional volume of $X_i$.