GAUSSIAN UNITARY ENSEMBLE: THE EIGENVALUE POINT PROCESS 1. m−POINT CORRELATION FUNCTIONS All information concerning the distribution of eigenvalues in any of the classical ensembles
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چکیده
All information concerning the distribution of eigenvalues in any of the classical ensembles (GOE, GUE, Wishart) is encoded in the joint densities, for which we have obtained explicit formulas. Unfortunately, getting information out of the joint density requires integration, and many of the integrals that arise cannot be evaluated in any nice closed form. Nevertheless, it is possible to show that as the size N of the ensemble becomes large, certain interesting functions of the eigenvalues – for instance, the maximum – have (after suitable re-centering and re-scaling) limit distributions. The unitary ensembles are easier to handle than the orthogonal ones, so we will limit our attention to these (at least for now). To be definite, we will focus on the GUE, for which the joint distribution of the eigenvalues λi , listed in random order, is
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