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We explore a new way of probing scattering of closed strings in $AdS_5\timesS^5$, which we call `the large $p$ limit'. It consists of studying four-pointcorrelators 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 thedual KK modes become short massive strings. In this regime the charge of thesingle-particle operators is order $\lambda^{1/4}$ and the dual KK modes are inbetween fields and strings. Starting from SUGRA we compute the large $p$ limitof the correlators by introducing an improved $AdS_5\times S^5$ Mellin spaceamplitude, and we show that the correlator is dominated by a saddle point. Ourresults are consistent with the picture of four geodesics shooting from theboundary of $AdS_5\times S^5$ towards a common bulk point, where they scatteras if they were in flat space. The Mandelstam invariants are put incorrespondence with the Mellin variables and in turn with certain combinationsof cross ratios. At the saddle point the dynamics of the correlator is directlyrelated 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 thefull type IIB S-matrix in the $AdS_5\times S^5$ Mellin amplitude, and how tostratify the latter in a large $p$ expansion. We compute the large $p$ limit ofall genus zero data currently available, pointing out additional hiddensimplicity of known results. We then show that the genus zero resummation atlarge $p$ naturally leads to the Gross-Mende phase for the minimal area surfacearound the bulk point. At one-loop, we first uncover a novel and finite Mellinamplitude, and then we show that the large $p$ limit beautifully asymptotes thegravitational S-matrix.

Large $p$ explorations. From SUGRA to big STRINGS in Mellin space