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Bulk universality for deformed GinUEs

Lu Zhang

Abstract

For the deformed complex Ginibre ensemble with a mean normal matrix, under certain assumptions on the mean matrix we prove that the same bulk statistics holds as in the complex Ginibre matrix bulk. This is the continuation of the previous joint papers ``Critical edge statistics for deformed GinUEs Preprint arXiv: 2311.13227v1'' and ``Repeated erfc statistics for deformed GinUEs Preprint arXiv: 2402.14362'', which deal with local eigenvalue statistics at the edge.

Bulk universality for deformed GinUEs

Abstract

For the deformed complex Ginibre ensemble with a mean normal matrix, under certain assumptions on the mean matrix we prove that the same bulk statistics holds as in the complex Ginibre matrix bulk. This is the continuation of the previous joint papers ``Critical edge statistics for deformed GinUEs Preprint arXiv: 2311.13227v1'' and ``Repeated erfc statistics for deformed GinUEs Preprint arXiv: 2402.14362'', which deal with local eigenvalue statistics at the edge.
Paper Structure (11 sections, 6 theorems, 153 equations)

This paper contains 11 sections, 6 theorems, 153 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Integral representation of correlation functions
  5. Notation
  6. Matrix variables
  7. Sketch of the proof for Theorem \ref{['2-complex-correlation']}
  8. Maximum Lemmas
  9. Proofs of Proposition \ref{['RNn delta']} and Theorem \ref{['2-complex-correlation']}
  10. Proof of Proposition \ref{['RNn delta']}
  11. Proof of Theorem \ref{['2-complex-correlation']}

Key Result

Theorem 1.2

For the ${\mathrm{GinUE}}_{N}(A_0)$ ensemble with $R_0\geq n$ and the assumptions A0 form on $A_0$, if $z_0$ is a bulk point which satisfies $a_{\alpha}\not=z_0$ for $\alpha=1,\cdots,t$ and $\sum_{\alpha=1}^t \frac{\tau c_{\alpha}}{|z_0-a_{\alpha}|^2+t_0}=1$ with $t_0>0$, then as $N\to \infty$ scale

Theorems & Definitions (12)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 2 more