# Laboratoire de Physique ThéoriqueToulouse - UMR 5152

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## Large Deviations of Extreme Eigenvalues of Random Matrices

David S. Dean and Satya N. Majumdar

par - 4 octobre 2006

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 () random matrix are positive (negative) decreases for large as where the parameter \beta characterizes the ensemble and the exponent is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number , thus generalizing the celebrated Wigner semi-circle law. The density of states generically exhibits an inverse square-root singularity at .