the TEK model corresponding to the NCYM, there is a constraint $a^2L=al=fixed$, which means that we take a scheme in which the noncommutative parameter $\theta$ is fixed. Then, both the continuum limit and the infinite volume limit are simultaneously taken (double scaling limit). Regardless of difference of the constraint, the scaling for the NCYM we should take is the same as that of the ordinary YM (\[EQ:one-loop\_scaling\]) by the nature of the logarithm scaling.

$\mathbb{Z}_N^4$ symmetry breaking in the TEK model {#SEC:breaking}
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As mentioned in the previous section, the $\mathbb{Z}_N^4$ symmetry breaking had not been observed in the older numerical simulation. However, there are several recent reports which indicate the symmetry breaking [@IO03; @Teper:2006sp; @Bietenholz:2006cz]. In [@Teper:2006sp], the symmetry breaking in the $D=4$ $SU(N)$ TEK model was studied in the case of the standard twist up to $N=144=12^2$. The authors of [@Teper:2006sp] performed the Monte-Carlo simulation starting both from a randomized configuration (“hot start”) and from the twist-eater solution (“cold start”). In both cases the $\mathbb{Z}_N^4$ symmetry begins to break at $N\ge 100=10^2$. At $N=144$ the symmetry breaking and restoration patterns they observed are $$\begin{aligned}
\begin{array}{cccccccccccl}
\mathbb{Z}_N^4 & \xrightarrow{\beta_c^H} & \mathbb{Z}_N^3 & \longrightarrow &
\mathbb{Z}_N^2 & \longrightarrow         & \mathbb{Z}_N^1 