Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials

We study the two-point correlation functions for the zeroes of systems of $SO(n+1)$-invariant Gaussian random polynomials on $\mathbb{RP}^n$ and systems of ${\rm isom}(\mathbb{R}^n)$-invariant Gaussian analytic functions. Our result reflects the same "repelling," "neutral," and "attracting" short-distance asymptotic behavior, depending on the dimension, as was discovered in the complex case by Bleher, Shiffman, and Zelditch. For systems of the ${\rm isom}(\mathbb{R}^n)$-invariant Gaussian analytic functions we also obtain a fast decay of correlations at long distances. We then prove that the correlation function for the ${\rm isom}(\mathbb{R}^n)$-invariant Gaussian analytic functions is "universal," describing the scaling limit of the correlation function for the restriction of systems of the $SO(k+1)$-invariant Gaussian random polynomials to any $n$-dimensional $C^2$ submanifold $M \subset \mathbb{RP}^k$. This provides a real counterpart to the universality results that were proved in the complex case by Bleher, Shiffman, and Zelditch. (Our techniques also apply to the complex case, proving a special case of the universality results of Bleher, Shiffman, and Zelditch.)