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Geodesics on the infinite dimensional symmetric space $\hcal$ of K\"ahlermetrics in a fixed K\"ahler class on a projective K\"ahler manifold X aresolutions of a homogeneous complex Monge-Amp\`ere equation in $X \times A$,where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parametersubgroups) on symmetric spaces $G_{\C}/G$. Donaldson, Arezzo-Tian andPhong-Sturm raised the question whether Monge-Amp\`ere geodesics can beapproximated by 1PS geodesics in the symmetric spaces of Bergman metrics.Phong-Sturm proved weak C^0 convergence of Bergman to Monge-Amp\`ere geodesicson a general \kahler manifold. In this article we prove convergence in $C^2(A\times X)$ in the case of toric K\"ahler metrics, extending our earlier resulton $\CP^1$.

Bergman metrics and geodesics in the space of K\"ahler metrics on toric varieties