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The smallest eigenvalue of $β$-Laguerre and $β$-Jacobi ensembles and multivariate orthogonal polynomials

Sungwoo Jeong

TL;DR

This work addresses finite-$n$ hard-edge statistics for the smallest eigenvalue in the $β$-Laguerre and $β$-Jacobi ensembles. It leverages Kaneko's integral formula to obtain exact, computable representations of the smallest-eigenvalue PDF and CDF in terms of scaled multivariate Laguerre and Jacobi polynomials evaluated at a diagonal argument, under integer parameter constraints. The authors derive new differentiation formulas for these multivariate polynomials and provide explicit rational Painlevé V (Laguerre) and Painlevé VI (Jacobi) solutions linked to the smallest eigenvalue, complemented by numerical experiments that validate the theory. These results yield explicit, finite-$n$ expressions that connect hard-edge statistics to multivariate orthogonal polynomials and Painlevé equations, enabling precise and efficient computation of extreme eigenvalue statistics.

Abstract

We study the smallest eigenvalue statistics of the $β$-Laguerre and $β$-Jacobi ensembles. Using Kaneko's integral formula, we show that the smallest eigenvalue marginal density and distribution functions of the two ensembles for any $β>0$ can be represented in terms of multivariate Laguerre and Jacobi polynomials evaluated at a multiple of the identity, provided that the exponent of $x$ in the Laguerre and Jacobi weights is an integer. These representations are readily computable in explicit form using existing symbolic algorithms for multivariate orthogonal polynomials. From these expressions, we derive new differentiation formulas for the multivariate Laguerre and Jacobi polynomials. Furthermore, we derive explicit solutions to the Painleve V and VI differential equations associated with the smallest eigenvalue of the LUE and JUE. We provide numerical experiments and examples.

The smallest eigenvalue of $β$-Laguerre and $β$-Jacobi ensembles and multivariate orthogonal polynomials

TL;DR

This work addresses finite- hard-edge statistics for the smallest eigenvalue in the -Laguerre and -Jacobi ensembles. It leverages Kaneko's integral formula to obtain exact, computable representations of the smallest-eigenvalue PDF and CDF in terms of scaled multivariate Laguerre and Jacobi polynomials evaluated at a diagonal argument, under integer parameter constraints. The authors derive new differentiation formulas for these multivariate polynomials and provide explicit rational Painlevé V (Laguerre) and Painlevé VI (Jacobi) solutions linked to the smallest eigenvalue, complemented by numerical experiments that validate the theory. These results yield explicit, finite- expressions that connect hard-edge statistics to multivariate orthogonal polynomials and Painlevé equations, enabling precise and efficient computation of extreme eigenvalue statistics.

Abstract

We study the smallest eigenvalue statistics of the -Laguerre and -Jacobi ensembles. Using Kaneko's integral formula, we show that the smallest eigenvalue marginal density and distribution functions of the two ensembles for any can be represented in terms of multivariate Laguerre and Jacobi polynomials evaluated at a multiple of the identity, provided that the exponent of in the Laguerre and Jacobi weights is an integer. These representations are readily computable in explicit form using existing symbolic algorithms for multivariate orthogonal polynomials. From these expressions, we derive new differentiation formulas for the multivariate Laguerre and Jacobi polynomials. Furthermore, we derive explicit solutions to the Painleve V and VI differential equations associated with the smallest eigenvalue of the LUE and JUE. We provide numerical experiments and examples.

Paper Structure

This paper contains 24 sections, 10 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Multivariate orthogonal polynomials
  4. Basic definitions
  5. Jack polynomials
  6. Hypergeometric function of a matrix argument
  7. Multivariate Laguerre polynomials
  8. Multivariate Jacobi polynomial
  9. Exact representations for the smallest eigenvalue of the L$\beta$E and J$\beta$E
  10. Matrix models for Laguerre and Jacobi ensembles
  11. The smallest eigenvalue of the Laguerre ensemble
  12. The smallest eigenvalue of the Jacobi ensemble
  13. Laguerre ensemble case
  14. Marginal density and distribution of the smallest eigenvalue of the Laguerre ensemble
  15. Differentiation formula and explicit Painlevé V solution
  16. ...and 9 more sections

Key Result

Lemma 3.1

For $n, \nu\in\mathbb{N}, \beta>0, \gamma>-1$, we have where the constant term is and $\tilde{\beta}, \tilde{\gamma}$ is defined in eq:dualparam.

Figures (4)

  • Figure 1: The empirical CDF (left) and PDF (right) of the smallest eigenvalue of the L$\beta$E, $\beta=\frac{5}{2}$, $n=4$, $\gamma=2$ are given in gray. The red curves represents Theorem \ref{['thm:laguerremainresult']}.
  • Figure 2: The empirical CDF (left) and PDF (right) of the smallest eigenvalue of the L$\beta$E, $\beta=e$, $n=3$, $\gamma=2$ are given in gray. The red curves represents Theorem \ref{['thm:laguerremainresult']}..
  • Figure 3: The empirical CDF (left) and PDF (right) of the smallest eigenvalue of the J$\beta$E, $\beta=4/3$, $n=3$, $\gamma_1 = 2$, $\gamma_2 = 2/3$ are given in gray. The red curves represents Theorem \ref{['thm:jacobimainresult']}.
  • Figure 4: The empirical CDF (left) and PDF (right) of the smallest eigenvalue of the J$\beta$E, $\beta=1/2$, $n = 2$, $\gamma_1=3$, $\gamma_2=1/4$ are given in gray. The red curves represents Theorem \ref{['thm:jacobimainresult']}.

Theorems & Definitions (11)

  • Definition 2.1
  • Lemma 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Corollary 3.3.1
  • Corollary 3.3.2
  • Lemma 4.1: Kaneko's integral formula
  • Proposition 4.2
  • Theorem 4.3
  • Corollary 4.3.1
  • ...and 1 more