  Restricted Isometry Property (RIP) is of fundamental importance in the theory
of compressed sensing and forms the base of many exact and robust recovery
guarantees in this field. A quantitative description of RIP involves bounding
the so-called RIP constants of measurement matrices. In this respect, it is
noteworthy that most results in the literature concerning RIP are upper bounds
of RIP constants, which can be interpreted as a theoretical guarantee of
successful sparse recovery. On the contrary, the land of lower bounds for RIP
constants remains uncultivated. Lower bounds of RIP constants, if exist, can be
interpreted as the fundamental limit aspect of successful sparse recovery. In
this paper, the lower bound of RIP constants Gaussian random matrices are
derived, along with a guide for generalization to sub-Gaussian random matrices.
This provides a new proof of the fundamental limit that the minimal number of
measurements needed to enforce the RIP of order $s$ is $\Omega(s\log({\rm
e}N/s))$, which is more straight-forward than the classical Gelfand width
argument. Furthermore, in the proof, we propose a useful technical tool
featuring the concentration phenomenon for top-$k$ sum of a sequence of i.i.d.
random variables, which is closely related to mainstream problems in statistics
and is of independent interest.
