  In this paper we revisit random linear under-determined systems with sparse
solutions. We consider $\ell_1$ optimization heuristic known to work very well
when used to solve these systems. A collection of fundamental results that
relate to its performance analysis in a statistical scenario is presented. We
start things off by recalling on now classical phase transition (PT) results
that we derived in \cite{StojnicCSetam09,StojnicUpper10}. As these represent
the so-called breaking point characterizations, we now complement them by
analyzing the behavior in a zone around the breaking points in a sense
typically used in the study of the large deviation properties (LDP) in the
classical probability theory. After providing a conceptual solution to these
problems we attack them on a "hardcore" mathematical level attempting/hoping to
be able to obtain explicit solutions as elegant as those we obtained in
\cite{StojnicCSetam09,StojnicUpper10} (this time around though, the final
characterizations were to be expected to be way more involved than in
\cite{StojnicCSetam09,StojnicUpper10}, simply, the ultimate goals are set much
higher and their achieving would provide a much richer collection of
information about the $\ell_1$'s behavior). Perhaps surprisingly, the final LDP
$\ell_1$ characterizations that we obtain happen to match the elegance of the
corresponding PT ones from \cite{StojnicCSetam09,StojnicUpper10}. Moreover, as
we have done in \cite{StojnicEquiv10}, here we also present a corresponding LDP
set of results that can be obtained through an alternative high-dimensional
geometry approach. Finally, we also prove that the two types of
characterizations, obtained through two substantially different mathematical
approaches, match as one would hope that they do.
