Collaborative Research: Theory and Algorithms for Beta Random Matrices: The Random Matrix Method of "Ghosts" and "Shadows"

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The main goal of the work described in this proposal is to introduce and perform a thorough theoretical and numerical analysis of the class of beta random matrix ensembles. The investigators will generalize and extend to any postive beta the classical real, complex, and quaternion random matrix ensembles which correspond to the cases 1, 2, 4, respectively. Already many results in infinite random matrix theory and a few results in finite random matrix theory suggest that the use of beta as a continuous parameter is reasonable. The key new concept in this proposal is the notion of a beta-random variable, an object which, for all practical purposes behaves as a beta-dimensional algebra over the reals. To develop the theory the PIs use the notions of "ghosts" and "shadows". A "ghost" is a beta-dimensional random variable and a "shadow" is a derived real or complex quantity that can be sampled. Along with the derivation of theoretical results, a major goal of this project is to provide algorithms for computation with these random matrix ensembles.A vast number of practical application ranging from bioinformatics, and genomics (population classification) to wireless communications (network capacity optimization) and military applications (automatic target classification) rely on the methods of multivariate statistics and in turn on random matrix theory. The proposed research will provide new algorithmic and theoretical tools for these applications as well as enable new applications and research directions in these fields.