Is there a critical order of the Abelian gauge theory in (2+1)D

In (2+1)D spacetime, it is known that the $U(1)$ gauge theory is always confined (according to Polyakov), while the $\mathbb{Z}_2$ gauge theory can support a deconfined phase. Now consider a generic Abelian gauge theory, i.e. the $\mathbb{Z}_n$ gauge theory (with $n$ being the order of the gauge group), the question is that does the $\mathbb{Z}_n$ theory in (2+1)D spacetime always support a deconfined phase, or there exist a critical $n_c$ such that for $n>n_c$ the $\mathbb{Z}_n$ theory is always confined?

The motivation of this question is that I believe in the $n\to\infty$ limit, the $\mathbb{Z}_n$ theory should behave the same as a $U(1)$ theory (let me know if it is not the case), while when $n=2$, it must reduce to the $\mathbb{Z}_2$ theory. But the $\mathbb{Z}_2$ theory is deconfined and the $U(1)$ theory is confined, so I am wondering if there exist a critical order $n_c$ such that beyond $n_c$ the Abelian gauge theory is always confined.

Also the gauge theory has a string correspondence (in terms of the dynamics of flux tubes, or in terms of the dynamics of membranes covering the Wilson loops). I would like to know if the above question could also be explained in the string theoretical language (in terms of the string tension, string renormalization ect.).

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Can you give some references which you are probably using for these? –  user6818 Nov 7 at 0:08
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