Random matrices and the average topology of the intersection of two quadrics

Abstract:

Let $X_{\mathbb R}$ be the zero locus in $\mathbb {R}\mathrm {P}^n$ of one or two independently and Kostlan distributed random real quadratic forms (this is equivalent to the corresponding symmetric matrices being in the Gaussian Orthogonal Ensemble). Denoting by $b(X_{\mathbb {R}})$ the sum of the Betti numbers of $X_{\mathbb {R}}$, we prove that \begin{equation}\lim _{n\to \infty }\frac {\mathbb {E}b(X_{\mathbb {R}})}{n}=1.\end{equation} The methods we use combine random matrix theory, integral geometry and spectral sequences: for one quadric hypersurface it is simply a corollary of Wigner’s semicircle law; for the intersection of two quadrics it is related to the (intrinsic) volume of the set of singular symmetric matrices of norm one.