Late time behavior of $n$-point spectral form factors in Airy and JT gravities

We study the late time behavior of $n$-point spectral form factors (SFFs) in two-dimensional Witten-Kontsevich topological gravity, which includes both Airy and JT gravities as special cases. This is conducted in the small $\mathrm{\hslash }$ expansion, where $\mathrm{\hslash }\sim e^{-1\mathrm{/}{G}_N}$ is the genus counting parameter and nonperturbative in Newton’s constant $G_N$. For one-point SFF, we study its absolute square at two different late times. We show that it decays by power law at $t\sim {\mathrm{\hslash }}^{-2\mathrm{/}3}$ while it decays exponentially at $t\sim {\mathrm{\hslash }}^{-1}$ due to the higher order corrections in $\mathrm{\hslash }$. We also study general $n(\ge 2)$-point SFFs at $t\sim {\mathrm{\hslash }}^{-1}$ in the leading order of the $\mathrm{\hslash }$ expansion. We find that they are characterized by a single function, which is essentially the connected two-point SFF and is determined by the classical eigenvalue density ${\rho }_0(E)$ of the dual matrix integral. These studies suggest that qualitative behaviors of $n$-point SFFs are similar in both Airy and JT gravities, where our analysis in the former case is based on exact results.