  Recently, we argued [Chin. Phys. Lett. $39$, 080502 (2022)] that the Ising
model simultaneously exhibits two upper critical dimensions $(d_c=4, d_p=6)$ in
the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we
perform a systematic study of the FK Ising model on hypercubic lattices with
spatial dimensions $d$ from 5 to 7, and on the complete graph. We provide a
detailed data analysis of the critical behaviors of a variety of quantities at
and near the critical points. Our results clearly show that many quantities
exhibit distinct critical phenomena for $4 < d < 6$ and $d\geq 6$, and thus
strongly support the argument that $6$ is also an upper critical dimension.
Moreover, for each studied dimension, we observe the existence of two
configuration sectors, two lengthscales, as well as two scaling windows, and
thus, two sets of critical exponents are needed to describe these behaviors.
Our finding enriches the understanding of the critical phenomena in the Ising
model.
