be important. The numerical computations in this work were in part carried out at the Yukawa Institute Computer Facility. The authors would like to thank Sinya Aoki, Hikaru Kawai, Jun Nishimura, Masanori Okawa, Yoshiaki Susaki, Hiroshi Suzuki and Shinichiro Yamato for stimulating discussions and comments. M. H. was supported by Special Postdoctoral Researchers Program at RIKEN. T. H. would like to thank the Japan Society for the Promotion of Science for financial support.

Double scaling limit as the noncommutative Yang-Mills theory {#sec:NCYM}
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The TEK model can be used to formulate gauge theories on noncommutative spaces nonperturbatively [@Aoki:1999vr; @Ambjorn:1999ts; @Connes:1997cr]. In this appendix, we give a review for the construction of the NCYM from the TEK model [@Aoki:1999vr], a discussion for the scaling and some supplemental comments for our analysis.

By taking $U_\mu=e^{iaA_\mu}$, where $a$ corresponds to the lattice spacing, and expanding the action of the TEK model (\[EQ:TEK\_action\]), we have its continuum version $$S_{TEK, continuum}=-\frac{1}{4g^2}\sum_{\mu\neq\nu}
         {\rm Tr}\left([A_\mu,A_\nu]-i\theta_{\mu\nu}\right)^2
\label{EQ:continuum_TEK}$$ up to higher order terms in $a$, where $$\theta_{\mu\nu}=\frac{2\pi n_{\mu\nu}}{Na^2},\quad
\frac{1}{2g^2}=a^4 \beta N.$$ Then, by expanding the action around a classical solution of (\[EQ:continuum\_TEK\]) $$A_\mu^{(0)}=\hat{p}_\mu, \qquad
[\hat{p}_\mu,\hat{p}_\nu]=i\theta_{\mu\nu},$$ we obtain the $U(1)$ NCYM on fuzzy ${\mathbb R}^4$ as follows. Let us define 