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

  title={The Potts-q random matrix model: loop equations, critical exponents, and rational case},
  author={Bertrand Eynard and G. Bonnet},
  journal={Physics Letters B},
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
Sums of Random Matrices and the Potts Model on Random Planar Maps
We compute the partition function of the $q$-states Potts model on a random planar lattice with $p\leq q$ allowed, equally weighted colours on a connected boundary. To this end, we employ its matrixExpand
Spanning Forests on Random Planar Lattices
The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q→0 limit, and extends the analogous notion for spanning trees, or dense self-avoidingExpand
Yang-Lee zeros of the two- and three-state Potts model defined on phi3 Feynman diagrams.
For the Ising model, an argument based on a symmetry of the saddle point equations leads to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. Expand
Boundary States of the Potts Model on Random Planar Maps
We revisit the 3-states Potts model on random planar triangulations as a Hermitian matrix model. As a novelty, we obtain an algebraic curve which encodes the partition function on the disc with bothExpand
Loop models on random maps via nested loops: the case of domain symmetry breaking and application to the Potts model
We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different localExpand
Nesting statistics in the $O(n)$ loop model on random planar maps
In the O(n) loop model on random planar maps, we study the depth – in terms of the number of levels of nesting – of the loop configuration, by means of analytic combinatorics. We focus on the "Expand
Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
We compute the generating functions of an model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, the topologies were already known, and here we compute all theExpand
On discrete surfaces: Enumerative geometry, matrix models and universality classes via topological recursion
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-calledExpand
Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits
We consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point of the model. The first part of this paperExpand
Counting Coloured Planar Maps: Differential Equations
We address the enumeration of q-coloured planar maps counted by the number of edges and the number of monochromatic edges. We prove that the associated generating function is differentiallyExpand


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
  • Phys. B466 (1996) 463-487, Nucl. Phys. B455 (1995) 577, B. Eynard, J. Zinn-Justin, Nucl. Phys. B386
  • 1992
  • Phys. B (Proc. Suppl.) 4
  • 1998
Nucl. Phys. B (Proc. Suppl.)
  • Nucl. Phys. B (Proc. Suppl.)
  • 1998
  • 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
  • Phys. B376
  • 1992