This talk is about joint work with Jian Song about spaces of Kahler metrics in a fixed Kahler class on a Kahler manifold M. There is a natural metric on this space which makes it an infinite dimensional symmetric space H (Mabuchi, Semmes, Donaldson). The geodesics in this space are important in relating GIT stability and existence of Calabi extremal metrics, and have been studied by Calabi, Chen, Donaldson, Arezzo-Tian, Phong-Sturm, and others. The geodesic equation is a Monge Ampere equation, and the geodesics are only known to be C^{1,1}. Phong-Sturm have shown that these infinite dimensional symmetric space geodesics can be uniformly approximated by the very simple geodesics on certain finite dimensional subspaces B_N of H known as Bergman metric spaces. My theme is that the geodesics of B_N approach those of H at least in C2 if M is a toric variety. The proof is based on lattice point sums and Bergman kernels, and is analogous to proofs of abscence of phase transitions in statistical mechanics.