  We consider the inverse problem of recovering the locations and amplitudes of
a collection of point sources represented as a discrete measure, given $M+1$ of
its noisy low-frequency Fourier coefficients. Super-resolution refers to a
stable recovery when the distance $\Delta$ between the two closest point
sources is less than $1/M$. We introduce a clumps model where the point sources
are closely spaced within several clumps. Under this assumption, we derive a
non-asymptotic lower bound for the minimum singular value of a Vandermonde
matrix whose nodes are determined by the point sources. Our estimate is given
as a weighted $\ell^2$ sum, where each term only depends on the configuration
of each individual clump. The main novelty is that our lower bound obtains an
exact dependence on the {\it Super-Resolution Factor} $SRF=(M\Delta)^{-1}$. As
noise level increases, the {\it sensitivity of the noise-space correlation
function in the MUSIC algorithm} degrades according to a power law in $SRF$
where the exponent depends on the cardinality of the largest clump. Numerical
experiments validate our theoretical bounds for the minimum singular value and
the sensitivity of MUSIC. We also provide lower and upper bounds for a min-max
error of super-resolution for the grid model, which in turn is closely related
to the minimum singular value of Vandermonde matrices.
