Moderate deviations for extreme eigenvalues of beta-Laguerre ensembles

Let lambda(max), lambda(min) be respectively the largest and smallest eigenvalues of beta-Laguerre ensembles with parameters (n, p, beta). For fixed beta > 0, under the condition that p is much larger than n(log n)(2), we obtain the full moderate deviation principles for lambda(max) and lambda(min) 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.