  Primordial black holes (PBHs) can be not only cold dark matter candidates but
also progenitors of binary black holes observed by LIGO-Virgo-KAGRA (LVK)
Collaboration. The PBH mass can be shifted to the heavy distribution if
multi-merger processes occur. In this work, we constrain the merger history of
PBH binaries using the gravitational wave events from the third
Gravitational-Wave Transient Catalog (GWTC-3). Considering four commonly used
PBH mass functions, namely the log-normal, power-law, broken power-law, and
critical collapse forms, we find that the multi-merger processes make a
subdominant contribution to the total merger rate. Therefore, the effect of
merger history can be safely ignored when estimating the merger rate of PBH
binaries. We also find that GWTC-3 is best fitted by the log-normal form among
the four PBH mass functions and confirm that the stellar-mass PBHs cannot
dominate cold dark matter.
