We compare the observed merger rate of galaxies over cosmic time and the frequency of collisional ring galaxies (CRGs), with analytic models and halo merger and collision rates from a large cosmological simulation. In the Lambda cold dark matter (LCDM) model we find that the cosmic {it merger fraction} does not evolve strongly between 0.2<z<2, implying that the observed decrease of the cosmic star formation rate since z~1 might not be tied to a disappearing population of major mergers. Halos hosting massive galaxies undergo on average ~2 mergers from z~2 up to present day, reflecting the late assembly time for the massive systems and the related downsizing problem. The cosmic {it merger rate} declines with redshift: at the present time it is a factor of 10 lower than at z~2, in reasonable agreement with the current available data. The rate of CRG formation derived from the interactions between halo progenitors up to z=2 is found to be a good tracer of the cosmic merger rate. In the LCDM model the rate of CRGs as well as the merger rate do not scale as (1+z)^m, as suggested by previous models. Our predictions of cosmic merger and CRG rates may be applied to forthcoming surveys such as GOODS and zCOSMOS.
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