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

The first two terms in the large $N$ asymptotic expansion of the $β$ moment of the characteristic polynomial for the Gaussian and Laguerre $β$-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.

No reviews yet.

No comments yet.