plotted in figure \[FIG:bc\_L\_mall\_skew1\]. From this figure we can find that the $\beta_c^L$ for each $L$ are reduced as we increase $m$, and the dependence is linear in $1/m$. The data at $1/m=0$ in this plot are linearly extrapolated values. An interesting point is the behavior for the case $L=5$. While the $\mathbb{Z}_N^4$ symmetry breaking is observed for $m=1, 2$ and $3$, it is not seen for $m=4$ because the $\beta_c^L$ reaches the bulk transition point $\beta_c^B\sim0.35$ by increasing $m$. Figure \[FIG:bc\_L\_mall\_skew2\] represents the same data in figure \[FIG:bc\_L\_mall\_skew1\], but the horizon axis is $L^2$. As we have seen in the $m=1$ case, the data for $L\geq8$ are well fitted by the linear function of $L^2$ for each $m$. From these figures, we find that the data for $L\geq8$ are well fitted globally by a function: $$\beta_c^L\sim0.0034L^2+\frac{0.060}{m}+0.19.
\label{EQ:bc_L_generic_skew}$$

  ------ ------- ------------------- ------- ------------------- ------- -------------------
  $L$    $N$         $\beta_c^L$     $N$         $\beta_c^L$     $N$         $\beta_c^L$
  $5$    $50$     $0.3525\pm0.0025$  $75$     $0.3475\pm0.0025$  -                -
  $6$    $72$     $0.3675\pm0.0025$  $108$    $0.3575\pm0.0025$  $144$    $0.3525\pm0.0025$
  $7$    $98$     $0.3925\pm0.0025$  $147$    $0.3875\pm0.0025$  $196$    $0.3825\pm0.0025$
  $8$    $128$    $0.4375\pm0.0025$  $192$    $0.4275\pm0.0025$  $256$    $0.4225\pm0.0025$
  $9$    $162$    $0.4925\pm0.0025$  $243$    $0.4825\pm0.0025$  $324$    $0.4775\pm0.0025$
  $10$   $200$    $0.5575\pm0.0025$  $300$    $0.5475\pm0.0025$  $400$    $0.5425\pm0.0025$
  ------ ------- ------------------- ------- ------------------- ------- 