Probabilistic Schubert Calculus: Asymptotics
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Probabilistic Schubert Calculus: Asymptotics Antonio Lerario1
· Leo Mathis1
Received: 4 March 2020 / Revised: 26 June 2020 / Accepted: 24 August 2020 © The Author(s) 2020
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 kplanes 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
8 = · 3π 5/2
π2 4
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 interesting integral over the space of polynomials that have all real roots. For instance: δ2,n
√ 9 3 = · 8n · n −3/2 1 + O n −1 . √ 2048 2π
Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for δ1,n involving a one-dimensional integral of certain combination of Elliptic functions.
B
Antonio Lerario [email protected] Leo Mathis [email protected]
1
SISSA, Trieste, Italy
123
A. Lerario, L. Mathis
1 Introduction 1.1 Random Real Enumerative Geometry In this paper we continue the study of real enumerative problems initiated in [9]. Our goal is to answer questions such as In average, how many lines intersect four random lines in RP3 ? To be more precise, let G(1, 3) be the Grassmannian of lines in RP3 . This is a homogeneous space equipped with a transitive action of the orthogonal group O(4) on it and there is a unique invariant probability measure defined on G(1, 3) invariant under this action. We fix L ∈ G(1, 3) and define the Schubert variety: (L) := { ∈ G(1, 3) | ∩ L = ∅} . Then the answer to the previous question is given by the number: δ1,3 := E# (g1 · (L) ∩ · · · ∩ g4 · (L)) where g1 , . . . , g4 are independent, taken uniformly at random from O(4) (with the normalized Haar measure). One can generalize to higher dimensions. Let G(k, n) be the Grassmannian of linear projective subspaces of dimension k in RPn . It is a homogeneous space with O(n + 1) acting transitively on it and with a unique O(n + 1)−invariant probability measure. We fix L ∈ G(n − k − 1, n) and introduce the corresponding Schubert variety: (L) := { ∈ G(k, n) | ∩ L = ∅} .
(1)
δk,n := E# g1 · (L) ∩ · · · ∩ g(k+1)(n−k) · (L)
(2)
We define
where g1 , . . . , g(k+1)(n−k) are independent, taken uniformly at random from O(n + 1) (with the normalized Haar measure). This number equals the average number of kdimensional subspaces of RPn meeting (k +1)(n −k) random subspaces of dimension (n − k − 1). 1.2 Previously on Probabilistic Schubert Calculus In the recent work [9], the
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