Probabilistic Schubert calculus

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree edegG.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 Rn, all of dimension n - k, where these subspaces are sampled uniformly and independently from G.n - k; n/. We express edegG.k; n/ in terms of the volume of an invariant convex body in the tangent space to the Grassmannian, and prove that for fixed k ≥ 2 and n → 1, (Formula Presented).