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Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge II: Level Densities

Folkmar Bornemann

TL;DR

The paper develops a unified, wave-function–based framework for Gaussian and Laguerre random matrix ensembles across β=1,2,4, focusing on soft-edge level densities. By expressing densities through skew-orthonormal and bi-orthonormal wave functions, it derives concise density formulas and establishes a precise soft-edge scaling with h_n ≍ n^{-2/3}, yielding Airy-based asymptotics and explicit correction terms. The authors prove an optimal second-order convergence rate via n' shifts, and they reconstruct generating-function expansions, providing both β=2 results and conjectural but supported first two corrections for β=1,4. The work tightly links density expansions to generating-function theory, confirms dualities between orthogonal and symplectic cases, and clarifies Laguerre-to-Gaussian scaling limits, with implications for edge statistics and precise large-n predictions in random matrix theory.

Abstract

We continue our work [arXiv:2403.07628] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre random matrix ensembles. By revisiting the construction of the associated skew-orthogonal polynomials in terms of wave functions, we obtain concise expressions for the level densities that are well suited for proving asymptotic expansions in powers of a certain parameter $h \asymp n^{-2/3}$. In the unitary case, the expansion for the level density can be used to reconstruct the first correction term in an established asymptotic expansion of the associated generating function. In the orthogonal and symplectic cases, we can even reconstruct the conjectured first and second correction terms.

Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge II: Level Densities

TL;DR

The paper develops a unified, wave-function–based framework for Gaussian and Laguerre random matrix ensembles across β=1,2,4, focusing on soft-edge level densities. By expressing densities through skew-orthonormal and bi-orthonormal wave functions, it derives concise density formulas and establishes a precise soft-edge scaling with h_n ≍ n^{-2/3}, yielding Airy-based asymptotics and explicit correction terms. The authors prove an optimal second-order convergence rate via n' shifts, and they reconstruct generating-function expansions, providing both β=2 results and conjectural but supported first two corrections for β=1,4. The work tightly links density expansions to generating-function theory, confirms dualities between orthogonal and symplectic cases, and clarifies Laguerre-to-Gaussian scaling limits, with implications for edge statistics and precise large-n predictions in random matrix theory.

Abstract

We continue our work [arXiv:2403.07628] on asymptotic expansions at the soft edge for the classical -dimensional Gaussian and Laguerre random matrix ensembles. By revisiting the construction of the associated skew-orthogonal polynomials in terms of wave functions, we obtain concise expressions for the level densities that are well suited for proving asymptotic expansions in powers of a certain parameter . In the unitary case, the expansion for the level density can be used to reconstruct the first correction term in an established asymptotic expansion of the associated generating function. In the orthogonal and symplectic cases, we can even reconstruct the conjectured first and second correction terms.

Paper Structure

This paper contains 28 sections, 11 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Concise expressions
  3. Scaling limits at the soft edge
  4. Convergence rates and first correction terms
  5. Asymptotic expansions
  6. Reconstruction of the expansion of the generating functions
  7. Matrix Ensembles and Skew-Orthonormal Wave Functions
  8. Parametrization of the matrix ensembles
  9. Index consistency rule
  10. Skew inner products
  11. Orthonormal wave functions
  12. Bi-orthonormal wave functions
  13. Skew-orthonormal wave functions
  14. Level Densities Written in Terms of Wave Functions
  15. Unitary ensembles
  16. ...and 13 more sections

Key Result

Theorem 2.1

With the definition $\psi_{-1}=0$, the bi-orthonormal system of wave functions satisfies, for $n=0,1,2,\ldots$, the differential and integral recursions

Figures (1)

  • Figure 1: Plots of the rescaled densities $\rho_{\beta,n,p}$ for $n=10$, $p=40$ (blue solid line) together with the asymptotic expansions \ref{['eq:omega']} truncated at $m=0,1,2$ (red dashed, orange dot-dashed, green dotted lines); the polynomial coefficients are taken from Theorems \ref{['thm:LUE']}/\ref{['thm:LOELSE']}. Left panel (a): LUE ($\beta=2$); middle panel (b): LOE ($\beta=1$); right panel (c): LSE ($\beta=4$).

Theorems & Definitions (27)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Remark 2.3
  • Corollary 2.2
  • proof
  • Remark 2.4
  • ...and 17 more