A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points

In this paper, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold $M$ with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, $E_{\lambda }(x,y)$, of the projection operator from $L^2(M)$ onto the direct sum of eigenspaces with eigenvalue smaller than ${\lambda }^2$ as $\lambda \to \mathrm{\infty }$. In the regime where $x,y$ are restricted to a compact neighborhood of the diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $E_{\lambda }$ and its derivatives of all orders, which generalizes a result of B'erard, who treated the on-diagonal case $E_{\lambda }(x,x)$. When $x,y$ avoid a compact neighborhood of the diagonal, we obtain this same improvement in an upper bound for $E_{\lambda }$. Our results imply that the rescaled covariance kernel of a monochromatic random wave locally converges in the $C^{\mathrm{\infty }}$ topology to a universal scaling limit at an inverse logarithmic rate.