Connecting Infinity to Soft Factors

In this note we study tree-level scattering amplitudes of gravitons under a natural deformation which in the large z limit can be interpreted either as a k-hard-particle limit or as a $(n-k)$-soft-particle limit. When $k=2$ this becomes the standard BCFW deformation while for $k=3$ it leads to the Risager deformation. The hard- to soft-limit map we define motivates a way of computing the leading order behavior of amplitudes for large z directly from soft limits. We check the proposal by applying the $k=3$ and $k=4$ versions to NMHV and N²MHV gravity amplitudes respectively. The former reproduces in a few lines the result recently obtained by using CHY-like techniques in {BCL}. The N²MHV formula is also remarkably simple and we give support for it using a CHY-like computation. In the $k=2$ case applied to any gravity amplitude, the multiple soft-limit analysis reproduces the correct O(z⁻²) behavior while explicitly showing the source of the mysterious cancellation among Feynman diagrams that tames the behavior from the O(zⁿ⁻⁵) of individual Feynman diagrams down to the O(z⁻²) of the amplitude.