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This article contains a detailed study, in the toric case, of the testconfiguration geodesic rays defined by Phong-Sturm. We show that the `Bergmanapproximations' of Phong-Sturm converge in C^1 to the geodesic ray and that thegeodesic ray itself is C^{1,1} and no better. The \kahler metrics associated tothe geodesic ray of potentials are discontinuous across certain hypersurfacesand are degenerate on certain open sets. A novelty in the analysis is theconnection between Bergman metrics, Bergman kernels and the theory of largedeviations.

Test configurations, large deviations and geodesic rays on toric varieties