Probabilistic Schubert calculus

Abstract

Abstract We initiate the study of average intersection theory in real Grassmannians. We define the expected degree edeg ⁡ G ⁢ ( k , n ) {\operatorname{edeg}G(k,n)} of the real Grassmannian G ⁢ ( k , n ) {G(k,n)} as the average number of real k-planes meeting nontrivially k ⁢ ( n - k ) {k(n-k)} random subspaces of ℝ n {\mathbb{R}^{n}} , all of dimension n - k {n-k} , where these subspaces are sampled uniformly and independently from G ⁢ ( n - k , n ) {G(n-k,n)} . We express edeg ⁡ G ⁢ ( k , n ) {\operatorname{edeg}G(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 {k\geq 2} and n → ∞ {n\to\infty} , edeg ⁡ G ⁢ ( k , n ) = deg ⁡ G ℂ ⁢ ( k , n ) 1 2 ⁢ ε k + o ⁢ ( 1 ) , \operatorname{edeg}G(k,n)=\deg G_{\mathbb{C}}(k,n)^{\frac{1}{2}\varepsilon_{k}% +o(1)}, where deg ⁡ G ℂ ⁢ ( k , n ) {\deg G_{\mathbb{C}}(k,n)} denotes the degree of the corresponding complex Grassmannian and ε k {\varepsilon_{k}} is monotonically decreasing with lim k → ∞ ⁡ ε k = 1 {\lim_{k\to\infty}\varepsilon_{k}=1} . In the case of the Grassmannian of lines, we prove the finer asymptotic edeg ⁡ G ⁢ ( 2 , n + 1 ) = 8 3 ⁢ π 5 / 2 ⁢ n ⁢ ( π 2 4 ) n ⁢ ( 1 + 𝒪 ⁢ ( n - 1 ) ) . \operatorname{edeg}G(2,n+1)=\frac{8}{3\pi^{5/2}\sqrt{n}}\biggl{(}\frac{\pi^{2}% }{4}\biggr{)}^{n}(1+\mathcal{O}(n^{-1})). 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 ⊆ ℝ ⁢ P n - 1 {X\subseteq\mathbb{R}\mathrm{P}^{n-1}} of dimension n - k - 1 {n-k-1} its Chow hypersurface Z ⁢ ( X ) ⊆ G ⁢ ( k , n ) {Z(X)\subseteq G(k,n)} , consisting of the k-planes A in ℝ n {\mathbb{R}^{n}} whose projectivization intersects X. Denoting N := k ⁢ ( n - k ) {N:=k(n-k)} , we show that 𝔼 ⁢ # ⁢ ( g 1 ⁢ Z ⁢ ( X 1 ) ∩ ⋯ ∩ g N ⁢ Z ⁢ ( X N ) ) = edeg ⁡ G ⁢ ( k , n ) ⋅ ∏ i = 1 N | X i | | ℝ ⁢ P m | , \mathbb{E}\#(g_{1}Z(X_{1})\cap\cdots\cap g_{N}Z(X_{N}))=\operatorname{edeg}G(k% ,n)\cdot\prod_{i=1}^{N}\frac{|X_{i}|}{|\mathbb{R}\mathrm{P}^{m}|}, where each X i {X_{i}} is of dimension m = n - k - 1 {m=n-k-1} , the expectation is taken with respect to independent uniformly distributed g 1 , … , g m ∈ O ⁢ ( n ) {g_{1},\dots,g_{m}\in O(n)} and | X i | {|X_{i}|} denotes the m-dimensional volume of X i {X_{i}} .