Possible large-N transitions for complex Wilson loop matrices

It is shown that a very simple multiplicative random complex matrix model generalizes the large-N phase structure found in the unitary case: A perturbative regime is joined to a non-perturbative regime at a point where the smoothness of some quantities breaks down. A generic complex Wilson loop matrix in a field theory admitting a 't Hooft planar limit could display a phase transition in that limit as nonlinear effects become dominating over linear ones.

[1]  M. Nowak,et al.  Large-Nc confinement and turbulence. , 2008, Physical Review Letters.

[2]  P. Olesen.  A linear equation for Wilson loops , 2007, 0712.0923.

[3]  T. Bohr.  Tunneling in two dimensional QCD , 2008 .

[4]  R. Narayanan,et al.  Universality of large N phase transitions in Wilson loop operators in two and three dimensions , 2007, 0711.4551.

[5]  Boris Dubrovin,et al.  On universality of critical behaviour in Hamiltonian PDEs , 2008, 0804.3790.

[6]  R. Narayanan,et al.  Infinite N phase transitions in continuum Wilson loop operators , 2006, hep-th/0601210.

[7]  O. Johnson.  Free Random Variables , 2004 .

[8]  R. Janik,et al.  Multiplying unitary random matrices—universality and spectral properties , 2003, math-ph/0312043.

[9]  J. Jurkiewicz,et al.  Infinite products of large random matrices and matrix-valued diffusion , 2003, math-ph/0304032.

[10]  A. D. Jackson,et al.  Products of random matrices. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Y. Makeenko.  Methods of contemporary gauge theory , 2002 .

[12]  A. Zee,et al.  “Single ring theorem” and the disk-annulus phase transition , 2001, cond-mat/0104072.

[13]  Y. Fyodorov,et al.  ALMOST HERMITIAN RANDOM MATRICES : CROSSOVER FROM WIGNER-DYSON TO GINIBRE EIGENVALUE STATISTICS , 1997, cond-mat/9703152.

[14]  Joel E. Cohen,et al.  Subadditivity, generalized products of random matrices and operations research , 1988 .

[15]  Steven A. Orszag,et al.  Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method , 1987 .

[16]  K. Vahala.  Handbook of stochastic methods for physics, chemistry and the natural sciences , 1986, IEEE Journal of Quantum Electronics.

[17]  P. Bougerol,et al.  Products of Random Matrices with Applications to Schrödinger Operators , 1985 .

[18]  Charles M. Newman,et al.  The Stability of Large Random Matrices and Their Products , 1984 .

[19]  The spectral density for two-dimensional continuum QCD , 1981 .

[20]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[21]  J. Ginibre.  Statistical Ensembles of Complex, Quaternion, and Real Matrices , 1965 .

[22]  H. Furstenberg.  Noncommuting random products , 1963 .