  Restricted Isometry Constants (RICs) provide a measure of how far from an
isometry a matrix can be when acting on sparse vectors. This, and related
quantities, provide a mechanism by which standard eigen-analysis can be applied
to topics relying on sparsity. RIC bounds have been presented for a variety of
random matrices and matrix dimension and sparsity ranges. We provide explicitly
formulae for RIC bounds, of n by N Gaussian matrices with sparsity k, in three
settings: a) n/N fixed and k/n approaching zero, b) k/n fixed and n/N
approaching zero, and c) n/N approaching zero with k/n decaying inverse
logrithmically in N/n; in these three settings the RICs a) decay to zero, b)
become unbounded (or approach inherent bounds), and c) approach a non-zero
constant. Implications of these results for RIC based analysis of compressed
sensing algorithms are presented.
