Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On The Eigenvalue Rigidity of the Jacobi Unitary Ensemble

Dan Dai, Chenhao Lu

Abstract

In this paper, we prove an optimal global rigidity estimate for the eigenvalues of the Jacobi unitary ensemble. Our approach begins by constructing a random measure defined through the eigenvalue counting function. We then prove its convergence to a Gaussian multiplicative chaos measure, which leads to the desired rigidity result. To establish this convergence, we apply a sufficient condition from Claeys et al. \cite{CFL2021} and conduct an asymptotic analysis of the related exponential moments.

On The Eigenvalue Rigidity of the Jacobi Unitary Ensemble

Abstract

In this paper, we prove an optimal global rigidity estimate for the eigenvalues of the Jacobi unitary ensemble. Our approach begins by constructing a random measure defined through the eigenvalue counting function. We then prove its convergence to a Gaussian multiplicative chaos measure, which leads to the desired rigidity result. To establish this convergence, we apply a sufficient condition from Claeys et al. \cite{CFL2021} and conduct an asymptotic analysis of the related exponential moments.

Paper Structure

This paper contains 27 sections, 22 theorems, 258 equations, 7 figures.

Table of Contents

  1. Introduction and Statement of Results
  2. Main Results
  3. Organization of the paper
  4. The Riemann-Hilbert Problems for the Hankel Determinants
  5. Steepest Descent Analysis for the RH Problem
  6. Normalization: $Y\mapsto T$
  7. Contour Deformation: $T\to S$
  8. The Global Parametrix
  9. Local Parametrices in the Merging Case
  10. Local Parametrices in the Edge Region
  11. The Model RH Problem
  12. Construction of the Local Parametrix near $1$
  13. The Small-norm RH Problem
  14. Asymptotics of Hankel Determinants
  15. Asymptotics of Hankel Determinants in the Separated Regime
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $h_N(x)$ be defined as in hN. For any $\varepsilon>0$, we have

Figures (7)

  • Figure 1: Regions $\Omega$, $\overline{\Omega }$ and jump contours of the RH problem for $S$ in Case (I)
  • Figure 2: Regions $\Omega$, $\overline{\Omega }$ and jump contours of the RH problem for $S$ in Case (II)
  • Figure 3: Regions $\Omega$, $\overline{\Omega }$ and jump contours of the RH problem for $S$ in Case (III)
  • Figure 4: The jump contours of the RH problem for $\Phi$
  • Figure 5: The jump contours of the RH problem for $\Xi$
  • ...and 2 more figures

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.2
  • proof
  • Lemma 3.6
  • proof
  • Lemma 3.13
  • ...and 36 more