---
author:
- |
    Tatsuo Azeyanagi$^a$, Masanori Hanada$^{b,c}$, Tomoyoshi Hirata$^{a}$ and Tomomi Ishikawa$^{d}$\
    Department of Physics, Kyoto University,\
    Kyoto 606-8502, Japan\
    Theoretical Physics Laboratory, RIKEN Nishina Center,\
    Wako, Saitama 351-0198, Japan\
    Department of Particle Physics, Weizmann Institute of Science,\
    Rehovot 76100, Israel\
    RIKEN BNL Research Center, Brookhaven National Laboratory,\
    Upton, New York 11973, USA\
    E-mail:
title: 'Phase structure of twisted Eguchi-Kawai model'
---

Introduction
============

The large-$N$ gauge theories provide fruitful features to both phenomenology and string theory. They are simplified in the large-$N$ limit while preserving essential features of QCD [@'t; @Hooft:1973jz]. Additionally, dimensional reductions of ten-dimensional ${\cal N}=1$ super Yang-Mills theory (matrix model) are expected to provide nonperturbative formulations of superstring theory [@Banks:1996vh; @Ishibashi:1996xs; @Dijkgraaf:1997vv; @Maldacena:1997re], and can also be regarded as effective actions of D-branes [@Witten:1995ex]. Furthermore, their twisted reduced versions, which we study in this article, can provide a nonperturbative formulation of the gauge theories on noncommutative spaces (NCYM) [@Aoki:1999vr; @Ambjorn:1999ts]. In order to study the nonperturbative nature of these theories, numerical simulations using lattice regularizations are quite efficient. (Non-lattice simulations are also applicable for the reduced models. See references [@Hanada:2007ti; @Anagnostopoulos:2007fw] for the recent progress.)

In the large-$N$ limit there is an equivalence between the 