this caused us to believe that the EK equivalence in the TEK model does hold throughout the whole coupling region.

Limiting procedure {#SEC:limiting_procedure}
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As is well known, the scaling of the YM lattice theory behaves as $\beta\sim\log a^{-1}$ around the weak coupling limit, where $a$ is the lattice spacing, and which is obtained by one-loop perturbative calculation of the renormalization group equation. If we wish to construct the TEK model which corresponds to the YM theory by the EK equivalence, the scaling of the TEK model should obey that of the YM theory. In the TEK model, the lattice size $L$ relates to $N$. (For the twist we consider in this article, the relation is $N=mL^2$.) Then, the YM system with fixed physical size $l=aL$ can be obtained by the scaling $$\beta\sim\log a^{-1}\sim\log N.
\label{EQ:one-loop_scaling}$$ In order to obtain the large-$N$ limit with infinite volume, we should increase $\beta$ slower than the scaling (\[EQ:one-loop\_scaling\]). If it is not the case, the system shrinks to a point.

In the case of the NCYM, the scaling near the weak coupling limit is essentially same as the YM theory, that is, $\beta\sim\log a^{-1}$. (See appendix \[sec:NCYM\].) But if we wish to make 