  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.)
