The Potts-q random matrix model: loop equations, critical exponents, and rational case

@article{Eynard1999ThePR,
  title={The Potts-q random matrix model: loop equations, critical exponents, and rational case},
  author={Bertrand Eynard and G. Bonnet},
  journal={Physics Letters B},
  year={1999},
  volume={463},
  pages={273-279}
}
Abstract In this article, we study the q -state Potts random matrix models extended to branched polymers, by the equations of motion method. We obtain a set of loop equations valid for any arbitrary value of q . We show that, for q=2−2 cos l r π ( l , r mutually prime integers with l r ), the resolvent satisfies an algebraic equation of degree 2 r −1 if l + r is odd and r −1 if l + r is even. This generalizes the presently-known cases of q =1,2,3. We then derive for any 0≤ q ≤4 the Potts- q… Expand
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References

SHOWING 1-10 OF 24 REFERENCES
Strings with discrete target space
We investigate the field theory of strings having as a target space an arbitrary discrete one- dimensional manifold. The existence of the continuum limit is guaranteed if the target space is a DynkinExpand
The planar approximation. II
The planar approximation is reconsidered. It is shown that a saddle point method is ineffective, due to the large number of degrees of freedom. The problem of eliminating angular variables isExpand
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
A handbook of mathematical functions that is designed to provide scientific investigations with a comprehensive and self-contained summary of the mathematical functions that arise in physical andExpand
B410 (93) 377
  • J. Ambjorn, G. Thorleifsson, M. Wexler, Nucl. Phys
  • 1995
Nucl
  • Phys. B466 (1996) 463-487, Nucl. Phys. B455 (1995) 577, B. Eynard, J. Zinn-Justin, Nucl. Phys. B386
  • 1992
Nucl
  • Phys. B (Proc. Suppl.) 4
  • 1998
Nucl. Phys. B (Proc. Suppl.)
  • Nucl. Phys. B (Proc. Suppl.)
  • 1998
Nucl
  • Phys. B410 (93) 377, J. Ambjorn, G. Thorleifsson, M. Wexler, Nucl. Phys. B439
  • 1995
Theory of random matrices in mesoscopic quantum physics
  • Theory of random matrices in mesoscopic quantum physics
  • 1994
Nucl
  • Phys. B376
  • 1992
...
1
2
3
...