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Computation of marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles

Peter J. Forrester, Santosh Kumar

Abstract

We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles. In the case $β=1$ this is a question of long standing in the mathematical statistics literature. A recursive procedure to accomplish this task is given for $β$ a positive integer, and the parameter $λ_1$ a non-negative integer. This case is special due to a finite basis of elementary functions, with coefficients which are polynomials. In the Laguerre case with $β= 1$ and $λ_1 + 1/2$ a non-negative integer some evidence is given of their again being a finite basis, now consisting of elementary functions and the error function multiplied by elementary functions. Moreover, from this the corresponding distributions in the fixed trace case permit a finite basis of power functions, as also for $λ_1$ a non-negative integer. The fixed trace case in this setting is relevant to quantum information theory and quantum transport problem, allowing particularly the exact determination of Landauer conductance distributions in a previously intractable parameter regime. Our findings also aid in analyzing zeros of the generating function for specific gap probabilities, supporting the validity of an associated large $N$ local central limit theorem.

Computation of marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles

Abstract

We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi ensembles. In the case this is a question of long standing in the mathematical statistics literature. A recursive procedure to accomplish this task is given for a positive integer, and the parameter a non-negative integer. This case is special due to a finite basis of elementary functions, with coefficients which are polynomials. In the Laguerre case with and a non-negative integer some evidence is given of their again being a finite basis, now consisting of elementary functions and the error function multiplied by elementary functions. Moreover, from this the corresponding distributions in the fixed trace case permit a finite basis of power functions, as also for a non-negative integer. The fixed trace case in this setting is relevant to quantum information theory and quantum transport problem, allowing particularly the exact determination of Landauer conductance distributions in a previously intractable parameter regime. Our findings also aid in analyzing zeros of the generating function for specific gap probabilities, supporting the validity of an associated large local central limit theorem.
Paper Structure (12 sections, 2 theorems, 75 equations, 6 figures)

This paper contains 12 sections, 2 theorems, 75 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The recursive scheme
  3. The marginal PDFs and inter-relations
  4. Differential-difference recurrences from the theory of Selberg integrals
  5. Recursive computation of $\{f_N(n;x) \}$ for $n \ge 2$
  6. Relationship to the eigenvalue density
  7. More on exact distributions in the Laguerre case with $\beta =1$
  8. The fixed trace variant
  9. Recursive computation of $F_N^{\rm L}(1;x)$ for $\lambda_1 + 1/2$ a non-negative integer
  10. Pfaffian form of $F_N^{\rm L}(1;x)$
  11. Functional form of $\{F_N^{\rm L}(n;x) \}$, $n \ge 2$
  12. Zeros of the generating function $\Xi_N(z;x)$

Key Result

Proposition 2

In the context of the real Wishart matrix $W_{n_1,N}^{(1)}$, set $\alpha=: (n_1-N-1)/2$. For $l$ a non-negative integer define Use this in the further definitions and from these quantities set For $N$ odd define too Further set $a_{i,i} = 0$, $a_{j,i} = - a_{i,j}$ for $j \ge i$ so in particular the matrix $[a_{i,j}]$ is anti-symmetric. Up to a known normalisation (which we omit for brevity ---

Figures (6)

  • Figure 1: Plots of (a) marginal PDFs $f_N^{\rm L}(n;x)$ and (b) marginal CDFs $F_N^{\rm L}(n;x)$ of eigenvalues for Laguerre case with $N=5, \beta=3, \lambda_1=6$. In both panels, the curves from right to left correspond to $n=1$ (largest) to $n=N$ (smallest) eigenvalues.
  • Figure 2: Plots of marginal PDFs $f_N^{\rm J}(n;x)$ and marginal CDFs $F_N^{\rm J}(n;x)$ for Jacobi ensemble. Panels (a) and (b) are for parameter choices $N=4,\beta=3,\lambda_1=-1/2,\lambda_2=2$, whereas (c) and (d) are for $N=4,\beta=3, \lambda_1=2, \lambda_2=3/4$. In all plots, the curves from right to left correspond to $n=1$ (largest) to $n=N$ (smallest) eigenvalues.
  • Figure 3: Plots of marginal PDFs $f_N^{\rm L}(n;x)$ and marginal CDFs $F_N^{\rm L}(n;x)$ for Laguerre ensemble. Panels (a) and (b) are for parameter choices $N=6,\beta=1,\lambda_1=3/2$, whereas (c) and (d) are for $N=5,\beta=3, \lambda_1=-1/2, \lambda_2=3/4$. In all plots, the curves from right to left correspond to $n=1$ (largest) to $n=N$ (smallest) eigenvalues.
  • Figure 4: Plots of marginal PDFs $f_N^{\rm fL}(n;x)$ and marginal CDFs $F_N^{\rm fL}(n;x)$ for fixed-trace Laguerre ensemble. Panels (a), (b) are for parameter choices $N=4,\beta=3,\lambda_1=1$; (c), (d) are for $N=4,\beta=1, \lambda_1=-1/2$, and (e), (f) are for $N=3, \beta=3, \lambda_1=5/2$. In all plots, the curves from right to left correspond to $n=1$ (largest) to $n=N$ (smallest) eigenvalues.
  • Figure 5: Plots of Landauer conductance PDF for number of channels (a) $N_1=2, N_2=4$, (b) $N_1=3, N_2=3$, and (c) $N_1=5, N_2=9$.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Remark 1
  • Proposition 2
  • Proposition 3
  • proof
  • Remark 4