Spectral density asymptotics for Gaussian and Laguerre $\beta$-ensembles in the exponentially small region

Abstract

The first two terms in the large $N$ asymptotic expansion of the $\beta$ moment of the characteristic polynomial for the Gaussian and Laguerre $\beta$-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms $o(1)$ . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.

19 pages

References in corpus (3)

• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• From Random Matrices to Stochastic Operators
• A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix

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