2019/12/17 by Lerario, Antonio, Mathis, Léo · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1912.08291
In the recent paper [arXiv:1612.06893] P. Bürgisser and A. Lerario 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 ℝ\textrmPn 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 \mathbbG(k,n) and, in the case k=1, they proved that: δ1,n= \frac83π5/2 ⋅ ((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))/(2048√(2π)) ⋅ 8n ⋅ n-3/2 ( 1+O(n-1)). Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and we give an explicit formula for δ1,n involving a one dimensional integral of certain combination of Elliptic functions.