Large Deviations of Extreme Eigenvalues of Random Matrices
- Laboratoire de Physique Theorique (UMR 5152 du CNRS), Universite Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 4 (France)
- Laboratoire de Physique Theorique et Modeles Statistiques (UMR 8626 du CNRS), Universite Paris-Sud, Batiment 100, 91405 Orsay Cedex (France)
We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (NxN) random matrix are positive (negative) decreases for large N as {approx}exp[-{beta}{theta}(0)N{sup 2}] where the parameter {beta} characterizes the ensemble and the exponent {theta}(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number {zeta}, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at {zeta}.
- OSTI ID:
- 20861003
- Journal Information:
- Physical Review Letters, Vol. 97, Issue 16; Other Information: DOI: 10.1103/PhysRevLett.97.160201; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 0031-9007
- Country of Publication:
- United States
- Language:
- English
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