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A note on the exact simulation of a random eigenvalue of a GUE matrix

Luc Devroye, Jad Hamdan

TL;DR

This work tackles exact, efficient sampling of a randomly chosen eigenvalue from the Gaussian Unitary Ensemble gue$(n)$, whose density is $\frac{1}{n}\sum_{k=0}^{n-1} \phi_k(x)^2$. The authors reduce the problem to drawing from squared Hermite densities $\phi_k^2$ by first selecting a random index $k$ and then sampling from $\phi_k^2$ using a rejection framework with explicit envelopes $h_n$ and a refined bound via van Veen’s Hermite representation. They introduce a linear-time baseline sampler and a refined sublinear-time algorithm achieving $\mathbb{E}[\text{time}] = O(n^{2/3})$ for sampling from $\phi_k^2$, and thus for an eigenvalue of gue$(n)$; simulations corroborate the density accuracy. The results provide provable runtime guarantees for exact sampling of GUE eigenvalues, with broader implications for sampling determinantal point processes in random matrix theory.

Abstract

We develop a simple algorithm to generate random variables described by densities equaling squared Hermite functions. As an application, we show how to generate a randomly chosen eigenvalue of a matrix from the Gaussian Unitary Ensemble ({\textsc{gue}}) in sub-linear expected time.

A note on the exact simulation of a random eigenvalue of a GUE matrix

TL;DR

This work tackles exact, efficient sampling of a randomly chosen eigenvalue from the Gaussian Unitary Ensemble gue, whose density is . The authors reduce the problem to drawing from squared Hermite densities by first selecting a random index and then sampling from using a rejection framework with explicit envelopes and a refined bound via van Veen’s Hermite representation. They introduce a linear-time baseline sampler and a refined sublinear-time algorithm achieving for sampling from , and thus for an eigenvalue of gue; simulations corroborate the density accuracy. The results provide provable runtime guarantees for exact sampling of GUE eigenvalues, with broader implications for sampling determinantal point processes in random matrix theory.

Abstract

We develop a simple algorithm to generate random variables described by densities equaling squared Hermite functions. As an application, we show how to generate a randomly chosen eigenvalue of a matrix from the Gaussian Unitary Ensemble ({\textsc{gue}}) in sub-linear expected time.
Paper Structure (9 sections, 5 theorems, 37 equations, 3 figures)

This paper contains 9 sections, 5 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The density of a random GUE eigenvalue
  3. Generating random variates with squared Hermite densities
  4. Notation and preliminaries on Hermite polynomials
  5. Generating random variates with density $h_n/\int h_n$
  6. A first eigenvalue algorithm that is linear in $n$
  7. Main algorithm and runtime analysis
  8. Appendix: Proof of Proposition \ref{['vverror']}.
  9. Appendix: Simulations

Key Result

Theorem 1

For any $n\in \mathbf{N}$, let $\phi_n^2$ be the square of the $n-$th Hermite function (defined in the previous section). Then Furthermore, where $B=(\pi+1)^2\sqrt{8(\pi+1)/3}$.

Figures (3)

  • Figure 1: The function $h_n$, defined piecewise. As in Algorithm 1 (section \ref{['bonangen']}), $p_1$, $p_2$ and $p_3$ denote the area under the curve in $[0,x_1)$, $[x_1,x_2)$ and $[x_2,\infty)$ respectively.
  • Figure 2: The function $\phi_{10}^2$ (bold) bounded from below and above by $(f_{10}-\epsilon^{-})_{+}$ and $(f_{10}+\epsilon^{+})$ (dashed).
  • Figure 3:

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 3
  • Proposition 4
  • proof
  • Lemma 5