Bertrand Eynard

Matrix eigenvalue model: Feynman graph technique for all genera

We present the diagrammatic technique for calculating the free energy of the matrix eigenvalue model (the model with arbitrary power β by the Vandermonde determinant) to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves).

Topological expansion for the 1-hermitian matrix model correlation functions

We rewrite the loop equations of the hermitian matrix model, in a way which allows to compute all the correlation functions, to all orders in the topological

Hermitian matrix model free energy: Feynman graph technique for all genera

1/N^2

MATRICES COUPLED IN A CHAIN : I. EIGENVALUE CORRELATIONS

expansion, as residues on an hyperelliptical curve. Those residues, can be represented diagrammaticaly as Feynmann graphs of a cubic interaction field theory on the curve.We rewrite the loop equations of the hermitian matrix model, in a way which involves no derivative with respect to the potential, we compute all the correlation functions, to all orders in the topological 1/N2 expansion, as residues on an hyperelliptical curve. Those residues, can be represented as Feynman graphs of a cubic field theory on the curve.

Breakdown of universality in multi-cut matrix models

We present the diagrammatic technique for calculating the free energy of the Hermitian one-matrix model to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves).

Torus Knots and Mirror Symmetry

The general correlation function for the eigenvalues of p complex Hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.

Holomorphic anomaly and matrix models

We solve the puzzle of the disagreement between orthogonal polynomials methods and mean-field calculations for random N×N matrices with a disconnected eigenvalue support. We show that the difference does not stem from a 2 symmetry breaking, but from the discreteness of the number of eigenvalues. This leads to additional terms (quasiperiodic in N) which must be added to the naive mean-field expressions. Our result invalidates the existence of a smooth topological large-N expansion and some postulated universality properties of correlators. We derive the large-N expansion of the free energy for the general two-cut case. From it we rederive by a direct and easy mean-field-like method the two-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.

Free energy topological expansion for the 2-matrix model

We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full

Duality, Biorthogonal Polynomials¶and Multi-Matrix Models

{{\rm Sl}(2, \mathbb {Z})}