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Rates of Bulk Convergence for Ensembles of Classical Compact Groups

Mengchun Cai

Abstract

This paper considers random matrices distributed according to Haar measure in different classical compact groups. Utilizing the determinantal point structures of their nontrivial eigenangles, with respect to the $L_1$-Wasserstein distance, we obtain the rate of convergence for different ensembles to the sine point process when the dimension of matrices $N$ is sufficiently large. Specifically, the rate is roughly of order $N^{-2}$ on the unitary group and of order $N^{-1}$ on the orthogonal group and the compact symplectic group.

Rates of Bulk Convergence for Ensembles of Classical Compact Groups

Abstract

This paper considers random matrices distributed according to Haar measure in different classical compact groups. Utilizing the determinantal point structures of their nontrivial eigenangles, with respect to the -Wasserstein distance, we obtain the rate of convergence for different ensembles to the sine point process when the dimension of matrices is sufficiently large. Specifically, the rate is roughly of order on the unitary group and of order on the orthogonal group and the compact symplectic group.

Paper Structure

This paper contains 6 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Outline of the Proof
  4. Decomposition of Operators
  5. CUE bulk
  6. Other ensembles

Key Result

Proposition 1.1

meckes_random_2019 For any $N\in\mathbb{Z}^+$, let The nontrivial eigenangles of uniformly distributed random matrices in any of $\mathbb{SO}_{2N},~\mathbb{SO}^-_{2N},~\mathbb{U}_N$ and $\mathbb{SP}_{2N}$ are a determinantal point process, with respect to the Lebesgue measure on $\Lambda$, with the following kernels $K_N$.

Theorems & Definitions (12)

  • Proposition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 2 more