Moderate deviations for extreme eigenvalues of beta-Laguerre ensembles

Abstract

Let [Formula: see text] be respectively the largest and smallest eigenvalues of beta-Laguerre ensembles with parameters [Formula: see text]. For fixed [Formula: see text], under the condition that [Formula: see text] is much larger than [Formula: see text], we obtain the full moderate deviation principles for [Formula: see text] and [Formula: see text] by using the asymptotic expansion technique. Interestingly, under this regime, our results show that asymptotically the exponential tails of the extreme eigenvalues are Gaussian-type distribution tail rather than the Tracy–Widom-type distribution tail.