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We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size   $k^{-\frac 14}$  . These improve the earlier results in the subject for smooth potentials, where an expansion exists in a   $k^{-\frac 12}$   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 [ 3 ]. 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 [ 2 ] (see Remark 3.5 of [ 2 ] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod   $O(e^{-k \delta } )$  .

Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials