Probabilistic Schubert Calculus: Asymptotics

  • Lerario, Antonio
  • Mathis, Léo
Publication date
January 2020
Publisher
Springer Science and Business Media LLC
Language
English

Abstract

In the recent paper Bürgisser and Lerario (Journal für die reine und angewandte Mathematik (Crelles J), 2016) introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by δk,n the average number of projective k-planes in RPn that intersect (k+1)(n−k) many random, independent and uniformly distributed linear projective subspaces of dimension n−k−1. They called δk,n the expected degree of the real Grassmannian G(k,n) and, in the case k=1, they proved that: δ1,n=83π5/2⋅(π24)n⋅n−1/2(1+O(n−1)). Here we generalize this result and prove that for every fixed integer k>0 and as n→∞, we have δk,n=ak⋅(bk)n⋅n−k(k+1)4(1+O(n−1)) where ak and bk are some (explicit) constants, and ak involves an interesti...

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