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Zeros of random functions generated with de Branges kernels

We study the point process given by the set of real zeros of random sums of orthonormal bases of reproducing kernels of de Branges spaces. Examples of these kernels are the cardinal sine, Airy and Bessel kernels. We find an explicit formula for the first intensity function in terms of the phase of the Hermite-Biehler function. We prove that the first intensity of the point process completely characterizes the underlying de Branges space. This result is a real version of the so called Calabi rigidity for GAFs proved by M. Sodin.