  In this paper we look at isometry properties of random matrices. During the
last decade these properties gained a lot attention in a field called
compressed sensing in first place due to their initial use in \cite{CRT,CT}.
Namely, in \cite{CRT,CT} these quantities were used as a critical tool in
providing a rigorous analysis of $\ell_1$ optimization's ability to solve an
under-determined system of linear equations with sparse solutions. In such a
framework a particular type of isometry, called restricted isometry, plays a
key role. One then typically introduces a couple of quantities, called upper
and lower restricted isometry constants to characterize the isometry properties
of random matrices. Those constants are then usually viewed as mathematical
objects of interest and their a precise characterization is desirable. The
first estimates of these quantities within compressed sensing were given in
\cite{CRT,CT}. As the need for precisely estimating them grew further a finer
improvements of these initial estimates were obtained in e.g.
\cite{BCTsharp09,BT10}. These are typically obtained through a combination of
union-bounding strategy and powerful tail estimates of extreme eigenvalues of
Wishart (Gaussian) matrices (see, e.g. \cite{Edelman88}). In this paper we
attempt to circumvent such an approach and provide an alternative way to obtain
similar estimates.
