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.

翻译：本文推导了张量Curie-Weiss Potts模型中经验磁化向量的极限分布以及自然参数的最大似然估计（ML估计）的相关结果。我们的结果揭示了令人惊讶的新相变现象，包括在参数平面内存在一条光滑曲线，在该曲线上磁化向量与ML估计具有混合极限分布（该分布由连续分量和离散分量共同构成），以及ML估计出现惊人的超高效现象——这规定了在某一类特定特殊点上，估计量以$N^{-3/4}$的速度收敛到某种非高斯分布，而在另一类不同特殊点上则以$N^{-5/6}$的速度收敛到另一种非高斯分布。最后一种情况仅可能出现在张量相互作用阶数与颜色数元组的某一特定取值时。这些结果进而被用于在能够实现一致估计的所有点上推导自然参数的渐近置信区间。