Associated to the Bergman kernels of a polarized toric \kahler manifold $(M, \omega, L, h)$ are sequences of measures $\{ \mu_k^z\}_{ k=1}^{ \infty}$ parametrized by the points $z \in M$. For each $z$ in the open orbit, we prove a central limit theorem for $\mu_k^z$. The center of mass of $\mu_k^z$ is the image of $z$ under the moment map; after re-centering at $0$ and dilating by $\sqrt{ k}$, the re-normalized measure tends to a centered Gaussian whose variance is the Hessian of the \kahler potential at $z$. We further give a remainder estimate of Berry-Esseen type. The sequence $\{ \mu_k^z\}$ is generally not a sequence of convolution powers and the proofs only involve \kahler analysis.
Categories: math.PR math.CV