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The merging rate of cosmic structures is computed, relying on the Ansatz thatthey can be predicted in the initial linear density field from the coalescenceof critical points with increasing smoothing scale, used here as a proxy forcosmic time. Beyond the mergers of peaks with saddle points (a proxy for halomergers), we consider the coalescence and nucleation of all sets of criticalpoints, including wall-saddle to filament-saddle and wall-saddle to minima (aproxy for filament and void mergers respectively), as they impact the geometryof galactic infall, and in particular filament disconnection. Analyticalpredictions of the one-point statistics are validated against multiscalemeasurements in 2D and 3D realisations of Gaussian random fields (thecorresponding code being available upon request) and compared qualitatively tocosmological $N$-body simulations at early times ($z\geq 10$) and large scales($\geq 5\, \mathrm{Mpc}/h$). The rate of filament coalescence is compared tothe merger rate of haloes and the two-point clustering of these events iscomputed, along with their cross-correlations with critical points. Thesecorrelations are qualitatively consistent with the preservation of theconnectivity of dark matter haloes, and the impact of the large scalestructures on assembly bias. The destruction rate of haloes and voids as afunction of mass and redshift is quantified down to $z=0$ for a $\Lambda$CDMcosmology. The one-point statistics in higher dimensions are also presented,together with consistency relations between critical point and critical eventcounts.

When do cosmic peaks, filaments or walls merge? A theory of critical events in a multi-scale landscape