Probabilistic Schubert Calculus: Asymptotics

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 delta _{k,n} the average number of projective k-planes in {mathbb {R}}mathrm {P}^n that intersect (k+1)(n-k) many random, independent and uniformly distributed linear projective subspaces of dimension n-k-1. They called delta _{k,n} the expected degree of the real Grassmannian {mathbb {G}}(k,n) and, in the case k=1, they proved that: δ1,n=83π5/2·π24n·n-1/21+On-1.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\delta _{1,n}= \\frac{8}{3\\pi ^{5/2}} \\cdot \\left( \\frac{\\pi ^2}{4}\\right) ^n \\cdot n^{-1/2} \\left( 1+{\\mathcal {O}}\\left( n^{-1}\\right) \\right) . \\end{aligned}$$\\end{document}Here we generalize this result and prove that for every fixed integer k>0 and as nrightarrow infty , we have δk,n=ak·bkn·n-k(k+1)41+O(n-1)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\delta _{k,n}=a_k \\cdot \\left( b_k\\right) ^n\\cdot n^{-\\frac{k(k+1)}{4}}\\left( 1+{\\mathcal {O}}(n^{-1})\\right) \\end{aligned}$$\\end{document}where a_k and b_k are some (explicit) constants, and a_k involves an interesting integral over the space of polynomials that have all real roots. For instance: δ2,n=9320482π·8n·n-3/21+On-1.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\delta _{2,n}= \\frac{9\\sqrt{3}}{2048\\sqrt{2\\pi }} \\cdot 8^n \\cdot n^{-3/2} \\left( 1+{\\mathcal {O}}\\left( n^{-1}\\right) \\right) . \\end{aligned}$$\\end{document}Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for delta _{1,n} involving a one-dimensional integral of certain combination of Elliptic functions.