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Asymptotic Expansions of the Limit Laws of Gaussian and Laguerre (Wishart) Ensembles at the Soft Edge

Folkmar Bornemann

TL;DR

The paper develops a comprehensive framework for asymptotic expansions of the soft-edge laws of Gaussian and Laguerre (Wishart) random-matrix ensembles across all three β-ensembles. By embedding Tracy–Widom limit laws into higher-order expansions with coefficients that are rational polynomials in the spectral-parameter $t$ (and Laguerre ratio $\tau$), the authors derive explicit expressions for the first several correction terms, proven for β=2 and hypothesized/algebraically supported for β=1,4. The approach hinges on a unified kernel-expansion strategy around the Airy kernel, turning-point analysis, and a linear-form/ self-consistent expansion framework that connects unitary to orthogonal and symplectic classes via Forrester–Rains interrelations. The results validate the expansions against large-scale simulations and illuminate the Gaussian-to-Laguerre transition and the role of parameter scaling in finite-size corrections, with broad implications for edge statistics in random-matrix theory and related Laguerre/Hermite polynomial asymptotics.

Abstract

The large-matrix limit laws of the rescaled largest eigenvalue of the orthogonal, unitary, and symplectic $n$-dimensional Gaussian ensembles -- and of the corresponding Laguerre ensembles (Wishart distributions) for various regimes of the parameter $α$ (degrees of freedom $p$) -- are known to be the Tracy-Widom distributions $F_β$ ($β=1,2,4$). We establish (paying particular attention to large or small ratios $p/n$) that, with careful choices of the rescaling constants and of the expansion parameter $h$, the limit laws embed into asymptotic expansions in powers of $h$, where $h \asymp n^{-2/3}$ resp. $h \asymp (n\,\wedge\,p)^{-2/3}$. We find explicit analytic expressions of the first few expansion terms as linear combinations of higher-order derivatives of the limit law $F_β$ with rational polynomial coefficients. The parametrizations are fine-tuned so that the expansion coefficients in the Gaussian cases are, for given $n$, the limits $p\to\infty$ of those of the Laguerre cases. Whereas the results for $β=2$ are presented with proof, the discussion of the cases $β=1,4$ is based on some hypotheses, focusing on the algebraic aspects of actually computing the polynomial coefficients. For the purposes of illustration and validation, the various results are checked against simulation data with large sample sizes.

Asymptotic Expansions of the Limit Laws of Gaussian and Laguerre (Wishart) Ensembles at the Soft Edge

TL;DR

The paper develops a comprehensive framework for asymptotic expansions of the soft-edge laws of Gaussian and Laguerre (Wishart) random-matrix ensembles across all three β-ensembles. By embedding Tracy–Widom limit laws into higher-order expansions with coefficients that are rational polynomials in the spectral-parameter (and Laguerre ratio ), the authors derive explicit expressions for the first several correction terms, proven for β=2 and hypothesized/algebraically supported for β=1,4. The approach hinges on a unified kernel-expansion strategy around the Airy kernel, turning-point analysis, and a linear-form/ self-consistent expansion framework that connects unitary to orthogonal and symplectic classes via Forrester–Rains interrelations. The results validate the expansions against large-scale simulations and illuminate the Gaussian-to-Laguerre transition and the role of parameter scaling in finite-size corrections, with broad implications for edge statistics in random-matrix theory and related Laguerre/Hermite polynomial asymptotics.

Abstract

The large-matrix limit laws of the rescaled largest eigenvalue of the orthogonal, unitary, and symplectic -dimensional Gaussian ensembles -- and of the corresponding Laguerre ensembles (Wishart distributions) for various regimes of the parameter (degrees of freedom ) -- are known to be the Tracy-Widom distributions (). We establish (paying particular attention to large or small ratios ) that, with careful choices of the rescaling constants and of the expansion parameter , the limit laws embed into asymptotic expansions in powers of , where resp. . We find explicit analytic expressions of the first few expansion terms as linear combinations of higher-order derivatives of the limit law with rational polynomial coefficients. The parametrizations are fine-tuned so that the expansion coefficients in the Gaussian cases are, for given , the limits of those of the Laguerre cases. Whereas the results for are presented with proof, the discussion of the cases is based on some hypotheses, focusing on the algebraic aspects of actually computing the polynomial coefficients. For the purposes of illustration and validation, the various results are checked against simulation data with large sample sizes.
Paper Structure (48 sections, 17 theorems, 404 equations, 8 figures)

This paper contains 48 sections, 17 theorems, 404 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Prior work
  4. The new findings of the paper
  5. Proof strategy for unitary ensembles
  6. The algebraic method for orthogonal and symplectic ensembles
  7. Expansion at the soft edge of the GUE
  8. Kernel expansion
  9. Simplifying transformation
  10. Proof of the general form of the expansion
  11. Functional form of the expansion terms
  12. Comments on prior work
  13. The case $m=1$
  14. The case $m=0$
  15. Expansion at the soft edge of the LUE and complex Wishart ensembles
  16. ...and 33 more sections

Key Result

Theorem 2.1

With the scaling and expansion parametersIn view of the later discussion, leading to eq:GUEnaturalScaling, these are, up to a possible rescaling of $h_n$, the unique 'natural' choices of parameters such that the leading order is normalized to $F_2(t)$ and the expansion proceeds in powers of $h_n\asy there holds the large matrix expansion of the $\mathop{\mathrm{\operator@font GUE}}\nolimits$ at th

Figures (8)

  • Figure 1: Plots of $E_{2,1}'(t)$ (left panel) and $E_{2,2}'(t)$ (middle panel). The right panel shows $E_{2,3}'(t)$ (black solid line) with the approximations \ref{['eq:F23']} for $n=10$ (red dotted line) and $n=80$ (green dashed line): the close match validates the functional forms displayed in \ref{['eq:F22']} and the differentiability of the expansion \ref{['eq:GUE']}. Details about the numerical method can be found in MR2895091MR2600548arxiv.2206.09411MR3647807.
  • Figure 2: [Expansion terms with a tilde are 'histogram adjusted', see Appendix \ref{['app:histo']}.] Left panel: histogram of the scaled largest eigenvalues $(\lambda_{\max} - \mu_n)/\sigma_n$, for $N=10^9$ draws from the $\mathop{\mathrm{\operator@font GUE}}\nolimits$ for matrix dimension $n=10$ (blue bars) vs. the Tracy--Widom limit density $F_2'$ (red dotted line) and its first finite-size correction $F_2' + h_n E_{2,1}'$ (black solid line). Middle panel: difference, scaled by $h_n^{-1}$, between the histogram midpoints and the limit density (blue bars) vs. the correction terms $E_{2,1}'$ (red dotted line) and $E_{2,1}' + h_n \tilde{E}_{2,2}'$ (black solid line). Right panel: difference, scaled by $h_n^{-2}$, between the histogram midpoints and the first finite-size correction (blue bars) vs. the correction terms $\tilde{E}_{2,2}'$ (red dotted line) and $\tilde{E}_{2,2}' + h_n \tilde{E}_{2,3}'$ (black solid line). Note that sampling errors are about to be visible here.
  • Figure 3: Plots of $E_{2,1;\tau}'(t)$ (left panel), $E_{2,2;\tau}'(t)$ (middle panel) and $E_{2,3;\tau}'(t)$ (right panel) for $\tau\in\{1, 8/9, 3/4, 5/9,1/3,1/9,0\}$. In the case $p\geqslant n$, these values of $\tau$ correspond to the ratios $p=n$, $p=4n$, $p=9n$, $p= 25 n$, $p\approx 100n$, $p\approx 1000 n$, $p=\infty$. See Figure \ref{['fig:LUESimulation']} for a comparison with sampling data in the case $p=4n$.
  • Figure 4: [Expansion terms with a tilde are 'histogram adjusted', see Appendix \ref{['app:histo']}.] Left panel: histogram of the scaled largest eigenvalues $(\lambda_{\max} - \mu_{n,p})/\sigma_{n,p}$, for $N=10^9$ draws, from the $\mathop{\mathrm{\operator@font LUE}}\nolimits$ with $n=10$, $p=40$, so that $\tau=\tau_{n,p}=8/9$ and $h=h_{n,p}=0.092482\cdots$ (blue bars) vs. the Tracy--Widom limit density $F_2'$ (red dotted line) and its second order correction $F_2' + h E_{2,1;\tau}' + h^2 \tilde{E}_{2,2;\tau}'$ (black solid line). Middle panel: difference, scaled by $h^{-1}$, between the histogram midpoints and the limit density (blue bars) vs. the correction terms $E_{2,1;\tau}'$ (red dotted line) and $E_{2,1;\tau}' + h \tilde{E}_{2,2;\tau}'$ (black solid line). Right panel: difference, scaled by $h^{-2}$, between the histogram midpoints and the first finite-size correction (blue bars) vs. the correction terms $\tilde{E}_{2,2;\tau}'$ (red dotted line) and $\tilde{E}_{2,2;\tau}' + h \tilde{E}_{2,3;\tau}'$ (black solid line).
  • Figure 5: Top row: GOE ($\beta=1$). Bottom row: GSE ($\beta=4$). Plots of $E_{\beta,1}'(t)$ (left panel) and $E_{\beta,2}'(t)$ (middle panel). The right panel shows $E_{\beta,3}'(t)$ (black solid line) with the approximations \ref{['eq:Fbeta3']} for $n=10$ (red dotted line) and $n=80$ (green dashed line): the close match validates the functional forms displayed in \ref{['eq:Fbeta2']} and the differentiability of the expansion \ref{['eq:GbetaE']}.
  • ...and 3 more figures

Theorems & Definitions (42)

  • Remark 1.1
  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • ...and 32 more