  In this paper, we derive results about the limiting distribution of the
empirical magnetization vector and the maximum likelihood (ML) estimates of the
natural parameters in the tensor Curie-Weiss Potts model. Our results reveal
surprisingly new phase transition phenomena including the existence of a smooth
curve in the interior of the parameter plane on which the magnetization vector
and the ML estimates have mixture limiting distributions, the latter comprising
of both continuous and discrete components, and a surprising superefficiency
phenomenon of the ML estimates, which stipulates an $N^{-3/4}$ rate of
convergence of the estimates to some non-Gaussian distribution at certain
special points of one type and an $N^{-5/6}$ rate of convergence to some other
non-Gaussian distribution at another special point of a different type. The
last case can arise only for one particular value of the tuple of the tensor
interaction order and the number of colors. These results are then used to
derive asymptotic confidence intervals for the natural parameters at all points
where consistent estimation is possible.
