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For a class of maximally degenerate families of Calabi-Yau hypersurfaces ofcomplex projective space, we study associated non-Archimedean and tropicalMonge-Amp\`ere equations, taking place on the associated Berkovich space, andthe essential skeleton therein, respectively. For a symmetric measure on theskeleton, we prove that the tropical equation admits a unique solution, up toan additive constant. Moreover, the solution to the non-Archimedean equationcan be derived from the tropical solution, and is the restriction of acontinuous semipositive toric metric on projective space. Together with thework of Yang Li, this implies the weak metric SYZ conjecture on the existenceof special Lagrangian fibrations in our setting.

Tropical and non-Archimedean Monge-Amp\`ere equations for a class of Calabi-Yau hypersurfaces