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Extreme eigenvalues of random matrices from Jacobi ensembles

B. Winn

Abstract

Two-term asymptotic formulae for the probability distribution functions for the smallest eigenvalue of the Jacobi $ β$-Ensembles are derived for matrices of large size in the régime where $ β> 0 $ is arbitrary and one of the model parameters $ α_1 $ is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases $ β= 2 $ and/or small values of $ α_1 $, explicit formulae involving more familiar functions, such as the modified Bessel function of the first kind, are presented.

Extreme eigenvalues of random matrices from Jacobi ensembles

Abstract

Two-term asymptotic formulae for the probability distribution functions for the smallest eigenvalue of the Jacobi -Ensembles are derived for matrices of large size in the régime where is arbitrary and one of the model parameters is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases and/or small values of , explicit formulae involving more familiar functions, such as the modified Bessel function of the first kind, are presented.
Paper Structure (23 sections, 20 theorems, 175 equations, 2 figures)

This paper contains 23 sections, 20 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Multi-variable hypergeometric functions and Jacobi polynomials
  3. Jack polynomials
  4. Multi-variable Jacobi polynomials
  5. Multivariate hypergeometric functions
  6. Some useful identities
  7. Calculations for finite-size matrices
  8. Probability distribution of the smallest eigenvalue
  9. Probability density of the smallest eigenvalue
  10. Two-term asymptotic formula
  11. Preliminary results
  12. The main contribution
  13. Bounding the tail terms
  14. Asymptotic formula
  15. Partial Differential Equation satisfied by $\fourIdx{}0{(\sigma)}1{F}$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\phi_1$ be the smallest eigenvalue of the $N\times N$ Jacobi $\beta$-Ensemble, $\beta>0$, with $\alpha_1\in{\mathbb N}_0$ and $\alpha_2>-1$. For $x>0$, The error estimate can depend on $\alpha_1, \alpha_2, \beta$ but is uniform for $x$ in a compact set.

Figures (2)

  • Figure 1: A plot of the empirical cumulative probability distribution (points) for the scaled smallest eigenvalue $N^2\phi_1$ for 1000 samples from the Jacobi Orthogonal Ensemble (i.e. $\beta=1$) with parameters $\alpha_1=0$, $\alpha_2=3$, for (a) $N=30$; (b) $N=1000$. Also plotted are the leading term and the $N=30$ first-order correction term of the prediction \ref{['eq:123']} (lines).
  • Figure 2: A plot of the empirical cumulative probability distribution (points) for the scaled smallest eigenvalue $N^2\phi_1$ for 1000 samples from the Jacobi $\beta$ Ensemble with $\beta=3$, with parameters $\alpha_1=2$, $\alpha_2=1.7$, for (a) $N=20$; (b) $N=1000$. Also plotted are the leading term and the $N=20$ first-order correction term of the prediction \ref{['eq:140']} (lines).

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 3.3
  • Corollary 3.4
  • Lemma 3.5
  • Theorem 4.1
  • Lemma 4.2
  • Corollary 4.3
  • ...and 10 more