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Large deviations for functions of two random projection matrices

Operator Algebras 2007-05-23 v1 Probability

Abstract

In this paper two independent and unitarily invariant projection matrices P(N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size NN converges to infinity. The result is formulated on the tracial state space TS(A)TS({\cal A}) of the universal CC^*-algebra A{\cal A} generated by two selfadjoint projections. The random pair (P(N),Q(N))(P(N),Q(N)) determines a random tracial state τNTS(A)\tau_N \in TS({\cal A}) and τN\tau_N satisfies the large deviation. The rate function is in close connection with Voiculescu's free entropy defined for pairs of projections.

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Cite

@article{arxiv.math/0504435,
  title  = {Large deviations for functions of two random projection matrices},
  author = {F. Hiai and D. Petz},
  journal= {arXiv preprint arXiv:math/0504435},
  year   = {2007}
}

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22 pages