Nonstandard Large and Moderate Deviations for the Laguerre Ensemble

TL;DR

This paper analyzes the weighted spectral measure of the Laguerre ensemble under nonstandard scaling with $\gamma_n\to\infty$ faster than $n$. Using the tridiagonal Laguerre model, it proves a Gaussian-type large deviation principle, a moderate deviation principle with a signed-measure limit, and a central limit theorem for polynomial test functions, all of which depend on the growth rate through parameters $\xi$ and $\zeta$. The results reveal a deep connection between Laguerre, Gaussian, and Jacobi ensembles in high-dimensional regimes, with the limiting behavior described by a semicircle-like law and shifts to signed measures in nonstandard scaling. The methodology combines LDP/MDP/CLT techniques via Jacobi coefficient analysis, the Dawson–Gärtner framework, and the Delta method, leveraging the Killip–Simon sum rule to characterize rate functions. Overall, the work clarifies how nonstandard scaling drives a crossover in spectral-measure limit theorems and highlights the role of signed-limit measures in moderate deviations and CLT settings.

Abstract

In this paper, we show limit theorems for the weighted spectral measure of the Laguerre ensemble under a nonstandard scaling, when the parameter grows faster than the matrix size. For this parameter scaling, the limit behavior is similar to the case of the Gaussian ensemble. We show a large deviation principle, moderate deviations and a CLT for the spectral measure. For the moderate deviations and the CLT, we observe a particular dependence on the rate of the parameter and a corrective shift by a signed measure. The proofs are based on the tridiagonal representation of the Laguerre ensemble.

Theorems & Definitions (14)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 4.1
• proof
• proof : Proof of Theorem \ref{['thm:LDPLaguerre']}
• Lemma 4.2
• proof
• Lemma 4.3
• proof