Mixed correlation function and spectral curve for the 2-matrix model

Abstract

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 spectral curve. In particular we prove the conjecture of M. Bertola, claiming that the spectral curve is the same curve which appears in the loop equations.

latex, 1 figure, 55 pages

References in corpus (21)

• 2D Gravity and Random Matrices
• A New Double-Scaling Limit of N=4 Super Yang-Mills Theory and PP-Wave Strings
• Free energy topological expansion for the 2-matrix model
• Matrices coupled in a chain. I. Eigenvalue correlations
• Duality, Biorthogonal Polynomials and Multi-Matrix Models
• An exact formula for general spectral correlation function of random Hermitian matrices
• Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem
• Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices
• Partition functions for Matrix Models and Isomonodromic Tau functions
• Pair Correlator in the Itzykson-Zuber Integral
• Ratios of characteristic polynomials in complex matrix models
• Mixed correlation functions in the 2-matrix model, and the Bethe ansatz
• Bilinear semi-classical moment functionals and their integral representation
• The Riemann-Hilbert Problem for the Bi-Orthogonal Polynomials
• 2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals
• Mixed Correlation Functions of the Two-Matrix Model
• The PDEs of biorthogonal polynomials arising in the two-matrix model
• Biorthogonal Polynomials for Potentials of two Variables and External Sources at the Denominator
• A short note about Morozov's formula
• The 2-matrix model, biorthogonal polynomials, Riemann-Hilbert problem, and algebraic geometry
• Orthogonal Polynomials for Potentials of two Variables with External Sources

Cited by in corpus (9)

• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
• Rational differential systems, loop equations, and application to the q-th reductions of KP
• The partition function of the two-matrix model as an isomonodromic tau-function
• Universality in the two matrix model with a monomial quartic and a general even polynomial potential
• Skew-orthogonal polynomials, differential systems and random matrix theory
• Average characteristic polynomials in the two-matrix model
• Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials
• Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis