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Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions

Longjun Zhan, Gordon Blower, Yang Chen, Mengkun Zhu

Abstract

In this paper, we study the probability density function, $\mathbb{P}(c,α,β, n)\,dc$, of the center of mass of the finite $n$ Jacobi unitary ensembles with parameters $α\,>-1$ and $β>-1$; that is the probability that ${\rm tr}M_n\in(c, c+dc),$ where $M_n$ are $n\times n$ matrices drawn from the unitary Jacobi ensembles. We first compute the exponential moment generating function of the linear statistics $\sum_{j=1}^{n}\,f(x_j):=\sum_{j=1}^{n}x_j,$ denoted by $\mathcal{M}_f(λ,α,β,n)$.

Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions

Abstract

In this paper, we study the probability density function, , of the center of mass of the finite Jacobi unitary ensembles with parameters and ; that is the probability that where are matrices drawn from the unitary Jacobi ensembles. We first compute the exponential moment generating function of the linear statistics denoted by .

Paper Structure

This paper contains 14 sections, 23 theorems, 169 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Toda Evolution and Riccati equations
  3. Painlevé V, Chazy Equation and discrete $\sigma$-form
  4. Painlevé V
  5. Chazy Equation
  6. The Discrete $\sigma-$form
  7. $\mathcal{M}_{f}(\lambda,\alpha,\beta,n)$ for large $n$ and finite $n$, Linear Statistics and the $\sigma$-form
  8. Log-concavity of the density of the center of mass
  9. Dyson's Coulomb Fluid
  10. Cumulants of the distribution of the center of mass
  11. Coefficients of $\sigma_n(-\lambda,\alpha,\beta)$ for $\alpha, \beta >-1$, $\alpha\neq0$, $\beta\neq0$
  12. The asymptotic expression of $\mathbb{P}(c,\alpha,\beta,n)$
  13. Uniform convexity
  14. Appendix A: On $\mathbb{P}(c,\alpha,\beta, n)$ for finite $n$.

Key Result

Proposition 2.1

The recursion coefficients $\alpha_n(\lambda)$ and $\beta_n(\lambda)$ satisfy the coupled Toda equations and the Toda molecule equation, see sogo1993,

Figures (3)

  • Figure 1: The coefficients distribution of $\mathbb{P}(c,0,0,n)$ and $\widehat{\mathbb{P}}(c,0,0,n)$
  • Figure 2: Compares $\mathbb{P}(c,1,2,n)$ with $\widehat{\mathbb{P}}(c,1,2,n)$
  • Figure 3: $\widehat{ \mathbb{P}}(c,0,0,n)$ changing with $n$

Theorems & Definitions (42)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 1
  • Remark 2
  • Remark 3
  • ...and 32 more