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Computing the Tracy-Widom Distribution for Arbitrary $β>0$

Thomas Trogdon, Yiting Zhang

TL;DR

This work generalizes the Tracy–Widom distribution to arbitrary $β>0$ by solving Bloemendal's boundary-value problem for the stochastic Airy operator and provides two robust numerical schemes—finite-difference and Fourier spectral—to compute $F_β(x)$ and its density with high accuracy. The authors deliver detailed stability analyses, error assessments, and timing benchmarks, and validate results against large random-matrix predictions, releasing a practical Julia package TracyWidomBeta.jl. By enabling precise computation of general-$β$ Tracy–Widom laws and offering extensive numerical results (including density evolution and higher-eigenvalue limits), the paper significantly advances both theoretical and applied aspects of random matrix theory. The methods yield accurate, adaptable tools for exploring β-generalized extremal eigenvalue statistics with broad impact in mathematical physics and statistics.

Abstract

We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $β$ is simply a parameter in the boundary-value problem, any $β> 0$ can be used, in principle. The limiting distribution of the $n$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.

Computing the Tracy-Widom Distribution for Arbitrary $β>0$

TL;DR

This work generalizes the Tracy–Widom distribution to arbitrary by solving Bloemendal's boundary-value problem for the stochastic Airy operator and provides two robust numerical schemes—finite-difference and Fourier spectral—to compute and its density with high accuracy. The authors deliver detailed stability analyses, error assessments, and timing benchmarks, and validate results against large random-matrix predictions, releasing a practical Julia package TracyWidomBeta.jl. By enabling precise computation of general- Tracy–Widom laws and offering extensive numerical results (including density evolution and higher-eigenvalue limits), the paper significantly advances both theoretical and applied aspects of random matrix theory. The methods yield accurate, adaptable tools for exploring β-generalized extremal eigenvalue statistics with broad impact in mathematical physics and statistics.

Abstract

We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since is simply a parameter in the boundary-value problem, any can be used, in principle. The limiting distribution of the th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.
Paper Structure (20 sections, 1 theorem, 57 equations, 22 figures, 3 tables, 3 algorithms)

This paper contains 20 sections, 1 theorem, 57 equations, 22 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Algorithm description
  3. Finite-difference discretization
  4. Spectral discretization
  5. Algorithm validation and comparison
  6. Eigenvalues of $\boldsymbol{\Delta x[T(\beta,\theta_{M},h)+U(\beta,x,\theta_{M},h)]}$ for finite-difference discretization
  7. Eigenvalues of $\boldsymbol{\Delta x(A+xB)}$ for the spectral discretization
  8. Error analysis
  9. Error across the domain
  10. Order of error
  11. Error with respect to $\boldsymbol{x_{0}}$
  12. Computation time
  13. Additional numerical results
  14. Comparison with large random matrices
  15. Evolution of the density $\boldsymbol{\rho(x,\theta):=\partial H(x,\theta)/\partial \theta}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

With probability one, for each $k\geq 0$, the set of eigenvalues of $\mathcal{H}_{\beta}$ has a well-defined $(k+1)$st lowest element $\Lambda_{k}$. Moreover, let $\lambda_{1}\geq \lambda_{2}\geq \cdots$ denote the eigenvalues of ${H}^{\beta}_{n}$. Then the vector converges in distribution to $(\Lambda_{0},\Lambda_{1},\dots,\Lambda_{k-1})$ as $n\to \infty$.

Figures (22)

  • Figure 1: The absolute stability region of trapezoidal method applied to the finite-difference discretization (the maroon-shaded region along with the orange boundary curve) and the eigenvalues of $\Delta x (T+U)$ (the red dots with white boundary) for $\beta=2$, $x=x_0,x_0-1,\dots,x_{N}$, $\Delta x=-10^{-3}$, and $M=10^3$.
  • Figure 2: The absolute stability region of BDF3 applied to the finite-difference discretization (the maroon-shaded region along with the orange boundary curve) and the eigenvalues of $\Delta x (T+U)$ (the red dots with white boundary) for $\beta=2$, $x=x_0,x_0-1,\dots,x_{N}$, $\Delta x=-10^{-3}$, and $M=10^3$.
  • Figure 3: The absolute stability region of BDF5 applied to the spectral discretization (the maroon-shaded region along with the orange boundary curve) and the eigenvalues of $\Delta x (A+xB)$ (the red dots with white boundary) for $\beta=2$, $x=x_0,x_0-1,\dots,x_{N}$, $\Delta x=-10^{-3}$, and $M=8\times 10^3$.
  • Figure 4: The absolute stability region of BDF6 applied to the spectral discretization (the maroon-shaded region along with the orange boundary curve) and the eigenvalues of $\Delta x (A+xB)$ (the red dots with white boundary) for $\beta=2$, $x=x_0,x_0-1,\dots,x_{N}$, $\Delta x=-10^{-3}$, and $M=8\times 10^3$.
  • Figure 5: Absolute errors from the two algorithms are presented over the domain $x\in [-10,13]$ for $\beta=1,2,4$ with $x_{0}=\lfloor 13/\sqrt{\beta}\rfloor$ and $\Delta x=-10^{-3}$. For the finite-difference discretization, $M=10^3$ is used, while for the spectral discretization, $M=8\times 10^3$ is employed.
  • ...and 17 more figures

Theorems & Definitions (4)

  • Theorem 1.1
  • Remark 3.1
  • Remark 3.2
  • Claim B.1