Let be a polarized toric Kahler manifold with polytope . Associated to this data is a family of probability measures on parametrized by They generalize the multi-nomial measures on the simplex, where and is the Fubini-Study measure. As is well-known, these measures satisfy a law of large numbers, a central limit theorem, a large deviations principle and entropy asymptotics. The measure of maximal entropy in this family corresponds to the center of mass of . All of these results generalize to any toric Kahler manifold, except the center of mass result, which holds for Fano toric Kahler-Einstein manifolds.
