×

zbMATH — the first resource for mathematics

Large deviations from the circular law. (English) Zbl 0916.60022
Let \(X^N\) be a random \(N \times N\) matrix whose entries are i.i.d. centered Gaussian random variables with variance \(N^{-1}\). Denote by \(\mu^N\) the empirical measure corresponding to the \(N\) complex eigenvalues of \(X^N\). The authors prove a full large deviation principle in the scale \(N^2\) for the distributions of \(\mu^N\), on the space of symmetric probability measures on C. (A probability measure \(\mu\) on C is symmetric if \(\mu(A^*) = \mu(A)\) where \(A^*\) is the complex conjugate set of \(A\).) The convex, good rate function is given explicitly and has a unique minimum at the circular law, i.e. at the uniform distribution on the unit disc. In particular, this yields an alternative derivation of the convergence of \(\mu^N\) to the circular law. The corresponding result for the self-adjoint case was proved by Ben Arous and Guionnet. The authors use similar techniques and make use of precise determinant computations due to Edelman and to Lehmann and Sommers.

MSC:
60F10 Large deviations
PDF BibTeX XML Cite
Full Text: DOI Link EuDML
References:
[1] BEN AROUS G., GUIONNET A., ( 1997). Large deviations for Wigner’s law and Voiculescu’s non commutative entropy, Prob. Th. Rel. Fields, 108, 517-542. Zbl0954.60029 MR1465640 · Zbl 0954.60029 · doi:10.1007/s004400050119
[2] EDELMAN A., ( 1997). The probability that a random real gaussian matrix has k real eigenvalues, related distributions, and the circular law, Jour. Multivariate Anal, 60, 203-232. Zbl0886.15024 MR1437734 · Zbl 0886.15024 · doi:10.1006/jmva.1996.1653
[3] GIRKO V.L., ( 1984). Circular law, Theory Prob. Appl. 29, 694-706. Zbl0565.60034 MR773436 · Zbl 0565.60034
[4] HlLLE E., ( 1962). Analytic function theory, Vol. II, Ginn and Co., Boston. Zbl0102.29401 MR201608 · Zbl 0102.29401
[5] LEHMANN N, SOMMERS H.J., ( 1991). Eigenvalue statistics of random real matrices, Phys. Rev. Let. 67. 941-944. Zbl0990.82528 MR1121461 · Zbl 0990.82528 · doi:10.1103/PhysRevLett.67.941
[6] MEHTA M.L., ( 1991). Random matrices, Academic Press, New York. Zbl0780.60014 MR1083764 · Zbl 0780.60014
[7] HIAI F., PETZ D., ( 1998). Logarithmic energy as entropy functional, in: Advances in differential equations and mathematical physics, E. Carlen, E, M. Harrell and M. Loss, editors, Contemporary Mathematics, vol. 217, 205-221, AMS, Providence. Zbl0893.15011 MR1606719 · Zbl 0893.15011
[8] HIAI F., PETZ D., ( 1997a). A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices, preprint. MR1743092 · Zbl 0954.60030
[9] HIAI F., PETZ D., ( 1997b). Eigenvalue density of the Wishart matrix and large deviations, preprint. MR1665279 · Zbl 0934.60006
[10] MHASKAR H.N., SAFF E.B., ( 1985). Where does the sup norm of a weighted polynomial live, Const. Approx., 1, 71-91. Zbl0582.41009 MR766096 · Zbl 0582.41009 · doi:10.1007/BF01890023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.