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

Details Spectral density asymptotics for Gaussian and Laguerre $β$-ensembles in the exponentially small region Spectral density asymptotics for Gaussian and Laguerre $β$-ensembles in the exponentially small region

2011-11-05

arxiv.org/abs/1111.1350v1

Physics

Mathematical Physics

19 pages

Scientific paper

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.

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