Rigid stationary determinantal processes in non-Archimedean fields

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Let F be a non-discrete non-Archimedean local field. For any subset S⊂ F with finite Haar measure, there is a stationary determinantal point process on F with correlation kernel 1ₛ(x-y), where 1ₛ is the Fourier transform of the indicator function 1ₛ. In this note, we give a geometrical condition on the subset S, such that the associated determinantal point process is rigid in the sense of Ghosh and Peres. Our geometrical condition is very different from the Euclidean case.