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We study the number of nodal domains in balls shrinking slightly above thePlanck scale for "generic" toral eigenfunctions. We prove that, up to thenatural scaling, the nodal domains count obeys the same asymptotic law as theglobal number of nodal domains. The proof, on one hand, uses new arithmeticinformation to refine Bourgain's de-randomisation technique at Planck scale.And on the other hand, it requires a Planck scale version of Yau's conjecturewhich we believe to be of independent interest.

Planck-scale number of nodal domains for toral eigenfunctions