Probabilistic Schubert Calculus

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree $\text{edeg}G(k,n)$ 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 $\text{edeg}G(k,n)$ 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 $n\to \mathrm{\infty }$, $$\text{edeg}G(k,n)=\text{deg}{G}_{\mathbb{C}}(k,n{)}^{\frac{1}{2}{ϵ}_k+o(1)},$$ where $$\text{deg}$ G_$\mathbb{C}$(k,n)$ denotes the degree of the corresponding complex Grassmannian and ${ϵ}_k$ is monotonically decreasing with $\lim_{k\to\infty} \epsilon_k = 1$. In the case of the Grassmannian of lines, we prove the finer asymptotic \begin{equation*} $\text{edeg}$ G(2,n+1) = \frac{8}{3\pi^{5/2}\sqrt{n}}, \left(\frac{\pi^2}{4} \right)^n \left(1+$\mathcal{O}$(n^{-1})\right). \end{equation*} The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set $X\subseteq \mathbb{R}{\text{P}}^{n-1}$ of dimension $n-k-1$ its Chow hypersurface $Z(X)\subseteq G(k,n)$, consisting of the $k$-planes $A$ in ${\mathbb{R}}^n$ whose projectivization intersects $X$. Denoting $N:=k(n-k)$, we show that $$\mathbb{E}\mathrm{\#}(g_{1}Z(X_1)\cap \cdots \cap g_{N}Z(X_N))=\text{edeg}G(k,n)\cdot \prod _{i=1}^N\frac{\mathrm{\mid }{X}_i\mathrm{\mid }}{\mathrm{\mid }\mathbb{R}{\text{P}}^m\mathrm{\mid }},$$ where each $X_i$ is of dimension $m=n-k-1$, the expectation is taken with respect to independent uniformly distributed $g_1,\dots ,g_m\in O(n)$ and $\mathrm{\mid }{X}_i\mathrm{\mid }$ denotes the $m$-dimensional volume of $X_i$.