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We study the two-point correlation functions for the zeroes of systems of   ${\rm SO}(n+1)$  -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. 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   ${\rm SO}(k+1)$  -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.

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