  We prove a new off-diagonal asymptotic of the Bergman kernels associated to
tensor powers of a positive line bundle on a compact K\"ahler manifold. We show
that if the K\"ahler potential is real analytic, then the Bergman kernel
accepts a complete asymptotic expansion in a neighborhood of the diagonal of
shrinking size $k^{-\frac14}$. These improve the earlier results in the subject
for smooth potentials, where an expansion exists in a $k^{-\frac12}$
neighborhood of the diagonal. We obtain our results by finding upper bounds of
the form $C^m m!^{2}$ for the Bergman coefficients $b_m(x, \bar y)$, which is
an interesting problem on its own. We find such upper bounds using the method
of Berman-Berndtsson-Sj\"ostrand. We also show that sharpening these upper
bounds would improve the rate of shrinking neighborhoods of the diagonal $x=y$
in our results. In the special case of metrics with local constant holomorphic
sectional curvatures, we obtain off-diagonal asymptotic in a fixed (as $k \to
\infty$) neighborhood of the diagonal, which recovers a result of Berman [Ber]
(see Remark 3.5 of [Ber] for higher dimensions). In this case, we also find an
explicit formula for the Bergman kernel mod $O(e^{-k \delta} )$.
