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

Abstract

Let (λ₁,…,λ_n) be β-Laguerre ensembles with parameters β,p,n and $\mu_{n}:=\sum\nolimits_{i=1}^{n}\delta_{X_{i}}$ with $X_{i}=(\lambda_{i}-\beta p)/(\beta\sqrt{np})$ for 1 ≤ i ≤ n. In this paper, as β varies and satisfies $\lim\nolimits_{n\rightarrow\infty}{\log n\over{\beta n}}=0$, we offer a modified Marchenko–Pastur law or semicircle law as the weak limits for the sequence μ_n when $\lim\nolimits_{n\rightarrow\infty}{n\over{p}}=\gamma\in\;(0,1]$ or $\lim\nolimits_{n\rightarrow\infty}{n\over{p}}=0$, respectively. This recovers some well-known results. Moreover, we give a full large deviation principle of μ_n with speed βn² and good rate function I_γ under the same condition to characterize the speed of the convergence. The minimizer of I_γ is a modified Marchenko–Pastur law for β ∈ (0,1] and the semicircle law for γ = 0.