Large Deviations of Extreme Eigenvalues of Random Matrices

Abstract

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[-βθ(0) N^2] where the parameter βcharacterizes the ensemble and the exponent θ(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 ζ, thus generalizing the celebrated Wigner semi-circle law. The density of states generically exhibits an inverse square-root singularity at ζ.

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

References in corpus (10)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• On Orthogonal and Symplectic Matrix Ensembles
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Critical points and supersymmetric vacua
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Cosmology From Random Multifield Potentials
• Exact Asymptotic Results for a Model of Sequence Alignment
• An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution
• Can we predict $\Lambda$ for the Non-SUSY sector of the Landscape ?

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