Combinatorial solution of the two-matrix model

Abstract

We write down and solve a closed set of Schwinger-Dyson equations for the twomatrix model in the large N limit. Our elementary method yields exact solutions for correlation functions involving angular degrees of freedom whose calculation was impossible with previously known techniques. The result sustains the hope that more complicated matrix models important for lattice string theory and QCD may also be solvable despite the problem of the angular integrations. As an application of our method we briefly discuss the calculation of wavefunctions with general matter boundary conditions for the Ising model coupled to 2D quantum gravity. Some novel insights into the relationship between lattice and continuum boundary conditions are obtained.

Noncommutative probability, matrix models, and quantum orbifold geometry

“…(11) that φ(z) has a branch cut between −1/(2γ) and 1/(2γ) but is otherwise holomorphic on the complex z-plane. The usual practice of matrix model analysis is to demand that the holomorphic structure of the generating function be as simple as possible [26,27,28,9,17,20]. Moving along the same spirit, we assert that ω 1 (z) is holomorphic on the whole complex plane except the same branch cut.…”

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

“…For the 2-matrix model, loop equations have been known since [33], and written in a more systematic way in [18][19][20]27]. …”

Rotational symmetry breaking in multimatrix models

“…U (N ) invariance, where N is the size of the matrix) which determines much of the universal behaviour in the large N limit. This global symmetry is present also in all the most relevant multi-matrix models (Ising model on random lattice [6,7], the Q-state Potts model [8,9,10,11,12,13], chain of matrices [14,15,16,17,18,19], models for coloring problem [20,21,22,23,24], vertex models [25,26,27,28,29], the meander model [30,31], the O(n)-model and some generalizations of it [32,33,34,35,36,37,38,39,40,41], and several others [42,43,44,45,46,47,48,49,50]. The list is not complete).…”