The purpose of the present paper is to set up a formalism inspired fromnon-Archimedean geometry to study K-stability. We first provide a detailedanalysis of Duistermaat-Heckman measures in the context of test configurations,characterizing in particular the trivial case. For any normal polarized variety(or, more generally, polarized pair in the sense of the Minimal Model Program),we introduce and study the non-Archimedean analogues of certain classicalfunctionals in K\"ahler geometry. These functionals are defined on the space oftest configurations, and the Donaldson-Futaki invariant is in particularinterpreted as the non-Archimedean version of the Mabuchi functional, up to anexplicit error term. Finally, we study in detail the relation between uniformK-stability and singularities of pairs, reproving and strengthening Y. Odaka'sresults in our formalism. This provides various examples of uniformly K-stablevarieties.
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