Unified Limits and Large Deviation Principles for β-Laguerre Ensembles in Global Regime.

Let (λ1,...,λn) be β-Laguerre ensembles with parameters β,p,n and with for 1 ≤ i ≤ n. In this paper, as β varies and satisfies , we offer a modified Marchenko–Pastur law or semicircle law as the weak limits for the sequence μn when or , respectively. This recovers some well-known results. Moreover, we give a full large deviation principle of μn with speed βn2 and good rate function Iγ under the same condition to characterize the speed of the convergence. The minimizer of Iγ is a modified Marchenko–Pastur law for β ∈ (0,1] and the semicircle law for γ = 0. [ABSTRACT FROM AUTHOR]