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).
Probabilistic Schubert calculus / Bürgisser, Peter; Lerario, Antonio. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - 760:(2020), pp. 1-58. [10.1515/crelle-2018-0009]
Probabilistic Schubert calculus
Lerario, Antonio
2020-01-01
Abstract
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).| File | Dimensione | Formato | |
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