We prove that the correlations functions, generated by the determinantal process of the Christoffel-Darboux kernel of an arbitrary order 2 ODE, do satisfy loop equations.
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)
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)
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)
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)
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)
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)