Much of classical random matrix theory deals with prescribing the joint distribution of the matrix entries, and then asking about the distribution of the resulting eigenvalues. In joint work with M. Meckes, we consider the opposite question: given a matrix whose eigenvalues are specified, but is otherwise random, what do the entries typically look like? More specifically, if a random matrix is chosen according to the canonical probability distribution on the set of real symmetric or complex Hermitian matrices having eigenvalues ?_1,…,?_n, then under mild conditions, when n is large linear functionals of the entries of such random matrices have approximately Gaussian joint distributions. In the context of quantum mechanics, these results can be viewed as describing the joint probability distribution of the expectation values of a family of observables on a quantum system in a random mixed state. I will also discuss other applications, in particular to the spectral distributions of submatrices, the classical invariant ensembles, and to a probabilistic counterpart of the Schur--Horn theorem, relating eigenvalues and diagonal entries of Hermitian matrices.
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