  The achievable and converse regions for sparse representation of white
Gaussian noise based on an overcomplete dictionary are derived in the limit of
large systems. Furthermore, the marginal distribution of such sparse
representations is also inferred. The results are obtained via the Replica
method which stems from statistical mechanics. A direct outcome of these
results is the introduction of sharp threshold for $\ell_{0}$-norm decoding in
noisy compressed sensing, and its mean-square error for underdetermined
Gaussian vector channels.
