We explore a new way of probing scattering of closed strings in $AdS_5times S^5$, which we call `the large $p$ limit. It consists of studying four-point correlators of single-particle operators in $mathcal{N}=4$ SYM at large $N$ and large t Hooft coupling $lambda$, by looking at the regime in which the dual KK modes become short massive strings. In this regime the charge of the single-particle operators is order $lambda^{1/4}$ and the dual KK modes are in between fields and strings. Starting from SUGRA we compute the large $p$ limit of the correlators by introducing an improved $AdS_5times S^5$ Mellin space amplitude, and we show that the correlator is dominated by a saddle point. Our results are consistent with the picture of four geodesics shooting from the boundary of $AdS_5times S^5$ towards a common bulk point, where they scatter as if they were in flat space. The Mandelstam invariants are put in correspondence with the Mellin variables and in turn with certain combinations of cross ratios. At the saddle point the dynamics of the correlator is directly related to the bulk Mellin amplitude, which in the process of taking large $p$ becomes the flat space ten-dimensional S-matrix. We thus learn how to embed the full type IIB S-matrix in the $AdS_5times S^5$ Mellin amplitude, and how to stratify the latter in a large $p$ expansion. We compute the large $p$ limit of all genus zero data currently available, pointing out additional hidden simplicity of known results. We then show that the genus zero resummation at large $p$ naturally leads to the Gross-Mende phase for the minimal area surface around the bulk point. At one-loop, we first uncover a novel and finite Mellin amplitude, and then we show that the large $p$ limit beautifully asymptotes the gravitational S-matrix.
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