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- M. C. BERGERE
- 2004

We construct biorthogonal polynomials for a measure over the complex plane which consists in the exponential of a potential −V (z, z *) and in a set of external sources at the numerator and at the denominator. We use the pseudonorm of these polynomials to calculate the resolvent integral for correlation functions of traces of powers of complex matrices… (More)

- M. C. Bergère
- 2008

For a restricted class of potentials (harmonic+Gaussian potentials), we express the resolvent integral for the correlation functions of simple traces of powers of complex matrices of size N , in term of a determinant; this determinant is function of four kernels constructed from the orthogonal polynomials corresponding to the potential and from their Cauchy… (More)

- M Bergère, B Eynard
- 2009

We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ∼ x p/q , the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p, q)-minimal models kernels. Those (p, q) kernels are written in terms of functions solutions of a linear… (More)

- M. Bergère
- 2006

We compute the mixed correlation function in a way which involves only the orthogonal polynomials with degrees close to n, (in some sense like the Christoffel Darboux theorem for non-mixed correlation functions). We also derive new representations for the differential systems satisfied by the biorthogonal polynomials, and we find new formulae for the… (More)

- M. C. BERGERE
- 2008

This publication is an exercise which extends to two variables the Christoffel's construction of orthogonal polynomials for potentials of one variable with external sources. We generalize the construction to biorthog-onal polynomials. We also introduce generalized Schur polynomials as a set of orthogonal, symmetric, non homogeneous polynomials of several… (More)

We write the loop equations for the β two-matrix model, and we propose a topological recursion algorithm to solve them, order by order in a small parameter. We find that to leading order, the spectral curve is a " quantum " spectral curve, i.e. it is given by a differential operator (instead of an algebraic equation for the hermitian case). Here, we study… (More)

- Michel Bergère, Ken-Ichiro Imura, Stéphane Ouvry
- 2001

We propose discrete TBA equations for models with discrete spectrum. We illustrate our construction on the Calogero-Moser model and determine the discrete 2-body TBA function which yields the exact N-body Calogero-Moser thermodynamics. We apply this algorithm to the Lieb-Liniger model in a harmonic well, a model which is relevant for the microscopic… (More)

- M. Bergère
- 2008

We find new representations for Itzykson-Zuber like angular integrals for arbitrary β, in particular for the orthogonal group O(n), the unitary group U(n) and the sym-plectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formula on n. The same methods gives also the Shatashvili's type… (More)