Limit Theorems and Phase Transitions in the Tensor Curie-Weiss Potts Model

TL;DR

This paper investigates the tensor Curie-Weiss Potts model, revealing new phase transition phenomena, including mixture distributions, super-efficiency of ML estimates, and deriving asymptotic confidence intervals for parameters.

Contribution

It introduces novel phase transition phenomena and super-efficiency results for ML estimates in the tensor Curie-Weiss Potts model, expanding understanding of its asymptotic behavior.

Findings

Existence of mixture limiting distributions on a smooth curve in parameter space.

Super-efficiency of ML estimates with non-Gaussian convergence rates.

Derivation of asymptotic confidence intervals for natural parameters.

Abstract

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.