Given an orthonormal basis {e_n} in a de Branges space (a certain Hilbert space of entire functions), and i.i.d. random variables {a_n} with real normal distribution N(0,1), we consider the Gaussian Analytic Function: F(z)=\sum_n a_n e_n(z). In this talk, we will present some results on the distribution of the real zeroes of F(z). It is natural that this random set contains information about the underline Hilbert space. We will see that this information is enough to characterize, up to some nice isomorphism, the de Branges space. We will also show some partial results on the hole probability. We will pay special attention to the Paley-Wiener space, and the spaces associated to the Airy and Bessel functions, because these three spaces also appear in point processes related to random matrices.This talk is based on a joint work with Jordi Marzo and Jan-Fredrik Olsen.