Probabilistic Schubert calculus

Abstract

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree{\operatorname{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 {\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\geq 2} and {n\to\infty},\operatorname{edeg}G(k,n)=\deg G_{\mathbb{C}}(k,n)^{\frac{1}{2}\varepsilon_{k}% +o(1)},where {\deg G_{\mathbb{C}}(k,n)} denotes the degree of the corresponding complex Grassmannian and {\varepsilon_{k}} is monotonically decreasing with {\lim_{k\to\infty}\varepsilon_{k}=1}. In the case of the Grassmannian of lines, we prove the finer asymptotic\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\subseteq\mathbb{R}\mathrm{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}\#(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}} 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 {|X_{i}|} denotes the m-dimensional volume of {X_{i}}.

On the Number of Real Zeros of Random Fewnomials

“…A.55]) implies the following result. (For a generalization to homogeneous spaces we refer to [14] and [9,Cor. A.3].)…”

Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

“…With probability one X is zero-dimensional and we can use integral geometry (see [14] or the appendix of [5]) to deduce that:…”

On the Number of Flats Tangent to Convex Hypersurfaces in Random Position

“…Probabilistic enumerative geometry. Recently, the second author of the current paper together with P. Bürgisser [4], have studied the similar problem of determining the average number of k-flats that simultaneously intersect d k,n many (n − k − 1)-flats in random position in RP n . They have called this number the expected degree of the real Grassmannian G(k, n), here denoted by δ k,n , and have claimed that this is the key quantity governing questions in random enumerative geometry of flats.…”

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Motivated by questions in real enumerative geometry [3,4,8,9,10,11,12] we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view. More precisely, we say that smooth convex hypersurfaces X 1 , .

…”

“…where δ k,n is the expected degree from [4] (the average number of k-flats incident to d k,n many random (n − k − 1)-flats), |Sch(k, n)| is the volume of the Special Schubert variety of k-flats meeting a fixed (n − k − 1)-flat (computed in [4]) and |Ω k (X)| is the volume of the manifold Ω k (X) ⊂ G(k, n) of all k-flats tangent to X. We give a formula for the evaluation of |Ω k (X)| in terms of some curvature integral of the embedding X ֒→ RP n and we relate it with the classical notion of intrinsic volumes of a convex set: |Ω k (∂C)| |Sch(k, n)| = 4|V n−k−1 (C)|, k = 0, .…”