Large Deviations of Extreme Eigenvalues of Random Matrices

Details Large Deviations of Extreme Eigenvalues of Random Matrices Large Deviations of Extreme Eigenvalues of Random Matrices

2006-09-26

arxiv.org/abs/cond-mat/0609651v2

Phys.Rev.Lett. 97 (2006) 160201

Physics

Condensed Matter

Statistical Mechanics

4 pages Revtex, 4 .eps figures included, typos corrected, published version

Scientific paper

10.1103/PhysRevLett.97.160201

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 (N\times N) random matrix are positive (negative) decreases for large N as \exp[-\beta \theta(0) N^2] where the parameter \beta characterizes the ensemble and the exponent \theta(0)=(\ln 3)/4=0.274653... 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 semi-circle law. The density of states generically exhibits an inverse square-root singularity at \zeta.

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