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We obtain explicit expressions for positive integer moments of theprobability density of eigenvalues of the Jacobi and Laguerre random matrixensembles, in the asymptotic regime of large dimension. These densities areclosely related to the Selberg and Selberg-like multidimensional integrals. Ourmethod of solution is combinatorial: it consists in the enumeration of certainclasses of lattice paths associated to the solution of recurrence relations.

Asymptotics of Selberg-like integrals by lattice path counting