  We explore the confinement-deconfinement phase transition (PT) of the first
order (FO) arising in $SU(N)$ pure Yang-Mills theory, based on Polyakov loop
models (PLMs), in light of the induced gravitational wave (GW) spectra. We
demonstrate that the PLMs with the Haar measure term, involving models
successful in QCD with $N=3$, are potentially incompatible with the large $N$
scaling for the thermodynamical quantities and the latent heat at around the
criticality of the FOPT reported from the lattice simulations. We then propose
a couple of models of polynomial form, which we call the 4-6 PLM (with four-
and six-point interactions among the basic PL fields which have center charge
1) and 4-8 PLM (with four- and eight-point interactions), and discuss how such
models can naturally arise in the presence of a heavy PL with charge 2. We show
that those models give the consistent thermodynamical and large $N$ properties
at around the criticality. The predicted GW spectra are shown to have high
enough sensitivity to be probed in the future prospected interferometers such
as LISA, BBO, DICIGO, and TianQin.
