  Recently, a hybrid percolation transitions (HPT) that exhibits both a
discontinuous transition and critical behavior at the same transition point has
been observed in diverse complex systems. In spite of considerable effort to
develop the theory of HPT, it is still incomplete, particularly when the
transition is induced by cluster merging dynamics. Here, we aim to develop a
theoretical framework of the HPT induced by such dynamics. We find that two
correlation-length exponents are necessary for characterizing the giant cluster
and finite clusters, respectively. Finite-size scaling method for the HPT is
also introduced. The conventional formula of the fractal dimension in terms of
the critical exponents is not valid. Neither the giant nor finite clusters are
fractals but they have fractal boundaries.
