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For any polarized variety (X,L), we show that test configurations and, moregenerally, R-test configurations (defined as finitely generated filtrations ofthe section ring) can be analyzed in terms of Fubini-Study functions on theBerkovich analytification of X with respect to the trivial absolute value onthe ground field. Building on non-Archimedean pluripotential theory, wedescribe the (Hausdorff) completion of the space of test configurations, withrespect to two natural pseudo-metrics, in terms of plurisubharmonic functionsand measures of finite energy on the Berkovich space. We also describe theHausdorff quotient of the space of all filtrations, and establish a 1--1correspondence between divisorial norms and divisorial measures, both beingdetermined in terms of finitely many divisorial valuations.

A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations