  We study the statistics of weak lensing convergence peaks, such as their
abundance and two-point correlation function (2PCF), for a wide range of
cosmological parameters $\Omega_m$ and $\sigma_8$ within the standard
$\Lambda$CDM paradigm, focusing on intermediate-height peaks with
signal-to-noise ratio (SNR) of $1.5$ to $3.5$. We find that the cosmology
dependence of the peak abundance can be described by a one-parameter fitting
formula that is accurate to within $\sim3\%$. The peak 2PCFs are shown to
feature a self-similar behaviour: if the peak separation is rescaled by the
mean inter-peak distance, catalogues with different minimum peak SNR values
have identical clustering, which suggests that the peak abundance and
clustering are closely interconnected. A simple fitting model for the rescaled
2PCF is given, which together with the peak abundance model above can predict
peak 2PCFs with an accuracy better than $\sim5\%$. The abundance and 2PCFs for
intermediate peaks have very different dependencies on $\Omega_m$ and
$\sigma_8$, implying that their combination can be used to break the degeneracy
between these two parameters.
