Then, because $\beta_c^L$ grows faster than the logarithm, the EK equivalence does not hold in the continuum limit.

Conclusions
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In order to study the nonperturbative nature of the large-$N$ gauge theory by lattice simulations, the large-$N$ reduction is very useful property for saving the computational effort. In this paper, we studied the phase structure of the TEK model, which has been a major way to realize the large-$N$ reduction. Contrary to the naive hope in old days, at least in ordinary twist prescriptions as investigated in this paper, the $\mathbb{Z}_N^4$ symmetry is broken even in the weak coupling region and hence a continuum limit as the planar gauge theory cannot be described by the TEK model. For the NCYM, the situation is the same. We can also consider a lot of variation for the twist prescription and the combination of reduced and non-reduced dimension. For example, in [@Guralnik:2002ru; @Bietenholz:2006cz], four-dimensional model with two commutative and two noncommutative directions was studied using two-dimensional lattice action. However, the $\mathbb{Z}_N^4$ symmetry is broken also in this model, and hence we cannot take a naive continuum limit.

Another way for the reduction is the QEK model, in which the eigenvalues of the link 