Probabilistic Schubert Calculus

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 $X_i$.