Entropy of Bergman measures of a toric Kaehler manifold

Associated to the Bergman kernels of a polarized toric Kaehler manifold $(M, \omega, L, h)$aresequencesofmeasures${\mu_k^z}_{k=1}^{\infty}$parametrizedbythepoints$z \in M$. We determine the asymptotics of the entropies $H({\mu }_k^z)$ of these measures. The sequence ${\mu }_k^z$ in some ways resembles a sequence of convolution powers; we determine precisely when it actually is such a sequence.