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 interesting integral over the space of polynomials that have all real roots. For instance:
δ2,n=93–√20482π−−√⋅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 give an explicit formula for δ1,n involving a one-dimensional integral of certain combination of Elliptic functions.
