Pierre van Moerbeke

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Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the cases that the eigenvalue probability(More)
2 Two-Toda lattice and reductions (Hänkel and Toeplitz) 17 2.1 Two-Toda on Moment Matrices and Identities for τ -Functions 18 2.2 Reduction to Hänkel matrices: the standard Toda lattice and a Virasoro algebra of constraints . . . . . . . . . . . . . . . . . 26 2.3 Reduction to Toeplitz matrices: two-Toda Lattice and an SL(2,Z)algebra of constraints . . . .(More)
into the algebras of skew-symmetric As and lower triangular (including the diagonal) matrices Ab (Borel matrices). We show that this splitting plays a prominent role also in the construction of the Toda symmetries and their action on τ−functions; it also plays a crucial role in obtaining the Virasoro constraints for matrix integrals and it ties up elegantly(More)
Random matrix theory has led to the discovery of novel matrix models and novel statistical distributions, which are defined by means of Fredholm determinants and which, in many cases, satisfy nonlinear ordinary or partial differential equations. A crucial observation is that these matrix integrals, upon appropriate deformation by means of exponentials(More)
Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA. Email: adler@math.brandeis.edu. The support of a National Science Foundation grant # DMS-01-00782 is gratefully acknowledged. Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA. E-mail:(More)
5 The action of Virasoro on Schur polynomials 31 Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA. Email: adler@brandeis.edu. The support of a National Science Foundation grant # DMS01-00782 is gratefully acknowledged. Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Clay Mathematics Institute, One(More)
Consider a semi-infinite skew-symmetric moment matrix, m ∞ evolving according to the vector fields ∂m/∂t k = Λ k m + mΛ ⊤k , where Λ is the shift matrix. Then The skew-Borel decomposition m ∞ := Q −1 JQ ⊤−1 leads to the so-called Pfaff Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the affine sym-plectic algebra. The(More)
Given the Hermitian, symmetric and symplectic ensembles, it is shown that the probability that the spectrum belongs to one or several intervals satisfies a nonlinear PDE. This is done for the three classical ensembles: Gaussian, Laguerre and Jacobi. For the Hermitian ensemble, the PDE (in the boundary points of the intervals) is related to the Toda lattice(More)