the skew-diagonal form, which is roughly three times different, by the factor in (\[EQ:potential\_difference\_generic\]).

Although the above crude estimation reproduces the linear $L^2$ behavior of $\beta_c^L$, we cannot explain the dependence on $m$. To catch the behavior completely, we need to make the discussion more sophisticated. However, we do not pursue this issue here because the $m$ dependence can be negligible at the larger $N$.

This argument can be applied for other twist prescriptions like taking the twist phase as $\exp(i\pi(L+1)/L)$, which is usually used for describing noncommutative spaces. (See appendix \[sec:NCYM\].)

Continuum and large-$N$ limit
-----------------------------

We have shown that the linear $L^2$ dependence of the critical point $\beta_c^L$ could be explained by the theoretical discussion in this section. While our simulation is restricted in the small $N$ region, we confirm that the behavior must continue to $N=\infty$ by combining with the discussion. Then the EK equivalence is valid only in the region $\beta >\beta_c^L\sim N$ even in the weak coupling limit and the large-$N$ limit. As we mentioned in the section \[SEC:limiting\_procedure\], both the ordinary YM with fixed physical volume and the NCYM theory with fixed noncommutative parameter have essentially logarithm scaling (\[EQ:one-loop\_scaling\]) near the weak coupling limit. 