  $\ell_1$ minimization can be used to recover sufficiently sparse unknown
signals from compressed linear measurements. In fact, exact thresholds on the
sparsity (the size of the support set), under which with high probability a
sparse signal can be recovered from i.i.d. Gaussian measurements, have been
computed and are referred to as "weak thresholds" \cite{D}. It was also known
that there is a tradeoff between the sparsity and the $\ell_1$ minimization
recovery stability. In this paper, we give a \emph{closed-form}
characterization for this tradeoff which we call the scaling law for
compressive sensing recovery stability. In a nutshell, we are able to show that
as the sparsity backs off $\varpi$ ($0<\varpi<1$) from the weak threshold of
$\ell_1$ recovery, the parameter for the recovery stability will scale as
$\frac{1}{\sqrt{1-\varpi}}$. Our result is based on a careful analysis through
the Grassmann angle framework for the Gaussian measurement matrix. We will
further discuss how this scaling law helps in analyzing the iterative
reweighted $\ell_1$ minimization algorithms. If the nonzero elements over the
signal support follow an amplitude probability density function (pdf)
$f(\cdot)$ whose $t$-th derivative $f^{t}(0) \neq 0$ for some integer $t \geq
0$, then a certain iterative reweighted $\ell_1$ minimization algorithm can be
analytically shown to lift the phase transition thresholds (weak thresholds) of
the plain $\ell_1$ minimization algorithm.
