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For a given polynomial $V(x)in mathbb C[x]$, a random matrix eigenvalues measure is a measure $prod_{1leq i<jleq N}(x_i-x_j)^2 prod_{i=1}^N e^{-V(x_i)}dx_i$ on $gamma^N$. Hermitian matrices have real eigenvalues $gamma=mathbb R$, which generalize to $gamma$ a complex Jordan arc, or actually a linear combination of homotopy classes of Jordan arcs, chosen such that integrals are absolutely convergent. Polynomial moments of such measure satisfy a set of linear equations called loop equations. We prove that every solution of loop equations are necessarily polynomial moments of some random matrix measure for some choice of arcs. There is an isomorphism between the homology space of integrable arcs and the set of solutions of loop equations. We also generalize this to a 2-matrix model and to the chain of matrices, and to cases where $V$ is not a polynomial but $V(x)in mathbb C(x)$.
In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e. the Stieltjes transform of the correlation functions). We obtain using Schwinger-Dyson equation (SDE) techniques the genera
In this paper we study the distribution of level crossings for the spectra of linear families A+lambda B, where A and B are square matrices independently chosen from some given Gaussian ensemble and lambda is a complex-valued parameter. We formulate
To any differential system $dPsi=PhiPsi$ where $Psi$ belongs to a Lie group (a fiber of a principal bundle) and $Phi$ is a Lie algebra $mathfrak g$ valued 1-form on a Riemann surface $Sigma$, is associated an infinite sequence of correlators $W_n$ th
The distribution of Bethe roots, solution of the inhomogeneous Bethe equations, which characterize the ground state of the periodic XXX Heisenberg spin-$frac{1}{2}$ chain is investigated. Numerical calculations shows that, for this state, the new inh
We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for