  We study K-stability properties of a smooth Fano variety X using
non-Archimedean geometry, specifically the Berkovich analytification of X with
respect to the trivial absolute value on the ground field. More precisely, we
view K-semistability and uniform K-stability as conditions on the space of
plurisubharmonic (psh) metrics on the anticanonical bundle of X. Using the
non-Archimedean Calabi-Yau theorem and the Legendre transform, this allows us
to give a new proof that K-stability is equivalent to Ding stability. By
choosing suitable psh metrics, we also recover the valuative criterion of
K-stability by Fujita and Li. Finally, we study the asymptotic Fubini-Study
operator, which associates a psh metric to any graded filtration (or norm) on
the anticanonical ring. Our results hold for arbitrary smooth polarized
varieties, and suitable adjoint/twisted notions of K-stability and Ding
stability. They do not rely on the Minimal Model Program.
