  The merging rate of cosmic structures is computed, relying on the Ansatz that
they can be predicted in the initial linear density field from the coalescence
of critical points with increasing smoothing scale, used here as a proxy for
cosmic time. Beyond the mergers of peaks with saddle points (a proxy for halo
mergers), we consider the coalescence and nucleation of all sets of critical
points, including wall-saddle to filament-saddle and wall-saddle to minima (a
proxy for filament and void mergers respectively), as they impact the geometry
of galactic infall, and in particular filament disconnection. Analytical
predictions of the one-point statistics are validated against multiscale
measurements in 2D and 3D realisations of Gaussian random fields (the
corresponding code being available upon request) and compared qualitatively to
cosmological $N$-body simulations at early times ($z\geq 10$) and large scales
($\geq 5\, \mathrm{Mpc}/h$). The rate of filament coalescence is compared to
the merger rate of haloes and the two-point clustering of these events is
computed, along with their cross-correlations with critical points. These
correlations are qualitatively consistent with the preservation of the
connectivity of dark matter haloes, and the impact of the large scale
structures on assembly bias. The destruction rate of haloes and voids as a
function of mass and redshift is quantified down to $z=0$ for a $\Lambda$CDM
cosmology. The one-point statistics in higher dimensions are also presented,
together with consistency relations between critical point and critical event
counts.
