Test configurations, large deviations and geodesic rays on toric varieties

This article contains a detailed study in the case of a toric variety of the geodesic rays ϕt defined by Phong and Sturm corresponding to test configurations T in the sense of Donaldson. We show that the ‘Bergman approximations’ ϕk(t,z) of Phong and Sturm converge in C1 to the geodesic ray ϕt, and that the geodesic ray itself is C1,1 and no better. In particular, the Kähler metrics associated to the geodesic ray of potentials are discontinuous across certain hypersurfaces and are degenerate on certain open sets.A novelty in the analysis is the connection between Bergman metrics, Bergman kernels and the theory of large deviations. We construct a sequence of measures on the polytope of the toric variety, show that they satisfy a large deviations principle, and relate the rate function to the geodesic ray.