[Paper Review] Large Deviations for Random Spectral Measures and Sum Rules

Fabrice Gamboa, Alain Rouault|arXiv (Cornell University)|Apr 28, 2008

Random Matrices and Applications20 references3 citations

TL;DR

This paper establishes a Large Deviation Principle (LDP) for random spectral measures associated with the GUE and its $β$-ensemble extensions, as well as Laguerre and Jacobi ensembles. The rate function comprises a reversed Kullback information term for the absolutely continuous part (relative to the semicircle or Marchenko-Pastur laws) and a singular part tied to extreme eigenvalue statistics, generalizing sum rule techniques from unitary to Hermitian matrix models.

ABSTRACT

We prove a Large Deviation Principle for the random spec- tral measure associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is a fixed unit vector (and more generally in the $\\beta$- extension of this model). The rate function consists of two parts. The contribution of the absolutely continuous part of the measure is the reversed Kullback information with respect to the semicircle distribution and the contribution of the singular part is connected to the rate function of the extreme eigenvalue in the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but in thoses cases the expression of the rate function is not so explicit.

Motivation & Objective

• To establish a Large Deviation Principle (LDP) for the spectral measure of the pair $(H_N, e)$, where $H_N$ is a GUE matrix and $e$ is a fixed unit vector.
• To extend the LDP to the $β$-Hermite, $β$-Laguerre, and $β$-Jacobi ensembles, generalizing the unitary case.
• To express the rate function as a sum of two components: reversed Kullback information for the absolutely continuous part and a term linked to extreme eigenvalue behavior.
• To explore connections between spectral measure LDPs and sum rules, particularly in the context of orthogonal polynomials and Verblunsky coefficients.
• To provide a framework for understanding the asymptotic behavior of weighted spectral measures in random matrix theory beyond empirical eigenvalue distributions.

Proposed method

• Derive the LDP for the spectral measure $µ_w^{(N)} = \sum_{k=1}^N \pi_k \delta_{\lambda_k}$ associated with GUE matrices and a fixed vector $e$, using the tridiagonal reduction of $H_N$.
• Use the fact that the weights $\pi_k = |\langle \psi_k, e \rangle|^2$ are distributed as Dirichlet random variables under the GUE, enabling explicit computation of the rate function.
• Apply the method of canonical moments (Verblunsky coefficients) and the Szegő formula to relate the spectral measure to a sum rule, which is key to deriving the rate function.
• For the $β$-ensembles, extend the LDP using the joint distribution of eigenvalues and weights, leveraging the Dufresne-Shepp formula and the $\beta$-Hermite ensemble's known joint density.
• Express the rate function as a sum of reversed Kullback information with respect to the limiting semicircle law and a correction term from the extreme eigenvalue rate function.
• For Laguerre and Jacobi ensembles, derive the LDP with an implicit rate function due to the lack of a known sum rule, though the structure is analogous.

Experimental results

Research questions

• RQ1 Does the weighted spectral measure $\mu_w^{(N)}$ of the GUE ensemble satisfy a Large Deviation Principle with speed $N$?
• RQ2 Can the rate function for the spectral measure in the GUE case be decomposed into a reversed Kullback information term and a term related to extreme eigenvalues?
• RQ3 How does the LDP for spectral measures in the $β$-Hermite ensemble compare to the classical GUE case, and what role does $\beta$ play?
• RQ4 To what extent can sum rules, such as the Szegő formula, be used to derive the rate function for spectral measures in random matrix ensembles?
• RQ5 Is it possible to generalize the LDP and rate function structure to the $β$-Laguerre and $β$-Jacobi ensembles, even when the rate function is not explicitly computable?

Key findings

• The spectral measure $\mu_w^{(N)}$ for the GUE ensemble satisfies an LDP with speed $N$, and the rate function is the sum of the reversed Kullback information with respect to the semicircle law and a term related to the rate function of the largest eigenvalue.
• For the $β$-Hermite ensemble, the rate function retains the same structure: reversed Kullback information with respect to the semicircle law plus a correction from extreme eigenvalue statistics.
• In the Laguerre and Jacobi ensembles, the LDP holds, but the rate function is not explicitly known due to the absence of a general sum rule, though the functional form is analogous.
• The method using Verblunsky coefficients and the Szegő formula successfully lifts the LDP from the canonical moments to the space of measures, providing a constructive way to derive the rate function.
• The rate function for the spectral measure in the GUE case is explicitly computed as $I(\nu) = \mathcal{K}(SC \,||\, \nu) + \mathcal{F}_{\text{extreme}}(\lambda_{\max})$, where $\mathcal{K}$ is the reversed Kullback divergence.
• The paper conjectures that for the Jacobi ensemble with parameters $\kappa_1 N, \kappa_2 N$, the rate function is $I(\nu) = \mathcal{K}(KMK \,||\, \nu) + \sum_j \mathcal{F}_J(E_j^\pm)$, generalizing the sum rule structure.