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On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities

Shuai-Xia Xu, Shu-Quan Zhao, Yu-Qiu Zhao

Abstract

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with $n$ discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the $n$ discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes $G$-function.

On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities

Abstract

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes -function.
Paper Structure (23 sections, 11 theorems, 147 equations, 1 figure)

This paper contains 23 sections, 11 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Model RH problem and the coupled Painlevé V system
  5. RH problem for the Fredholm determinant
  6. Model RH problem
  7. Lax pair and the coupled Painlevé V system
  8. Differential identity
  9. Asymptotic Analysis as $t$ $\rightarrow$ $0^+$
  10. Local parametrix
  11. Ratio RH problem
  12. Asymptotic behavior as $t\rightarrow0^+$
  13. Asymptotic Analysis as $t$ $\rightarrow$ $+\infty$
  14. Normalization
  15. Opening lenses and deformation
  16. ...and 8 more sections

Key Result

THEOREM 1

Let $\alpha>-\frac{1}{2}$, $\beta \in i\mathbb{R}$, $n \geq 1$, $\vec{\gamma}=\left(\gamma_0, \gamma_1, \dots, \gamma_{n-1}\right) \in[0,1)^n$ and $\vec{r}=\left(r_0, r_1, \dots, r_n\right) \in \mathbb{R}^{n+1}$ be such that $r_0< \dots <r_m=0< \dots <r_n$, we have the following integral expression where the Hamiltonian $H$ is defined by (hamiltonform) and (defhamilton) subject to the following a

Figures (1)

  • Figure 2: Jump contours and regions for $S$

Theorems & Definitions (18)

  • THEOREM 1
  • REMARK 1
  • THEOREM 2
  • COROLLARY 1
  • COROLLARY 2
  • COROLLARY 3
  • proof
  • COROLLARY 4
  • PROPOSITION 1
  • proof
  • ...and 8 more