Off-diagonal decay of toric Bergman kernels

We study the off-diagonal decay of Bergman kernels $\Pi_{h^k}(z,w)$ and Berezin kernels $P_{h^k}(z,w)$ for ample invariant line bundles over compact toric projective \kahler manifolds of dimension $m$. When the metric is real analytic, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$ where $D(z,w)$ is the diastasis. When the metric is only $C^{\infty}$ this asymptotic cannot hold for all $(z,w)$ since the diastasis is not even defined for all $(z,w)$ close to the diagonal. We prove that for general $C^{\infty}$ metrics, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$ as long as $w$ lies on the ${\mathbb R}_+^m$-orbit of $z$, and for general $(z,w)$, $\limsup_{k \to \infty} \frac{1}{k} \log P_{h^k}(z,w) \leq - D(z^*,w^*)$ where $D(z, w^*)$ is the diastasis between $z$ and the translate of $w$ by $(S^1)^m$ to the ${\mathbb R}_+^m$ orbit of $z$, complementary to Mike Christ's negative results (arXiv:1308.5644).