Rigidity of random stationary measures and applications to point processes

The mass rigidity of a stationary random measure M on Rᵈ or Zᵈ (called number rigidity for a point process) entails that for a bounded set A the knowledge of M on Ac determines M (A) ; the k-order rigidity for an integer k means we can recover the moments of the restriction of M on A^c up to order k. We show that the k-rigidity properties of a random stationary measure M can be characterised by the integrability properties around 0 of s, the continuous part of the spectral measure, by exploiting a connection with Schwartz' Paley-Wiener theorem for analytic functions of exponential type. If 1/s is not integrable in zero, M is mass rigid, and similarly if s has a zero of order 2k in 0, then M is k-rigid. In the continuous setting, these local conditions are also necessary if s has finitely many zeros, or is isotropic, or is at the opposite separable. This explains why no model seems to exhibit rigidity in dimension d 3, and allows to efficiently recover many recent rigidity results about point processes. In the discrete setting, these results hold provided \#A > 2k. For a continuous Determinantal point process with reduced kernel, k-rigidity is equivalent to (1-²) ^-1 having a zero of order 2k in 0, which answers questions on completeness and number rigidity. We also explore the consequences of these statements in the less tractable realm of Riesz gases.