Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Repeated erfc statistics for deformed GinUEs

Dang-Zheng Liu, Lu Zhang

Abstract

For an additive perturbation of the complex Ginibre ensemble under a deterministic matrix $X_0$, under certain assumption on $X_0$, we observe that there are only two kinds of local statistical patterns at the spectral edge: GinUE statistics and critical statistics, which corresponds to regular or quadratic vanishing spectral points. As a continuation of our previous study on critical statistics "Critical edge statistics for deformed GinUEs"(arXiv: 2311.13227), in this paper we establish the local statistics of GinUE type at the regular spectral edge, which is characterized by a repeated erfc integral found in "Phase transition of eigenvalues in deformed Ginibre ensembles"(arXiv: 2204.13171v2).

Repeated erfc statistics for deformed GinUEs

Abstract

For an additive perturbation of the complex Ginibre ensemble under a deterministic matrix , under certain assumption on , we observe that there are only two kinds of local statistical patterns at the spectral edge: GinUE statistics and critical statistics, which corresponds to regular or quadratic vanishing spectral points. As a continuation of our previous study on critical statistics "Critical edge statistics for deformed GinUEs"(arXiv: 2311.13227), in this paper we establish the local statistics of GinUE type at the regular spectral edge, which is characterized by a repeated erfc integral found in "Phase transition of eigenvalues in deformed Ginibre ensembles"(arXiv: 2204.13171v2).
Paper Structure (11 sections, 8 theorems, 180 equations)

This paper contains 11 sections, 8 theorems, 180 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Proof of Theorem \ref{['2-complex-correlation']}
  5. Notation and matrix variables
  6. Concentration reduction
  7. Proof of Theorem \ref{['2-complex-correlation']}: $z_0\neq 0$
  8. Proofs of Propositions \ref{['fTaylor']}, \ref{['gTaylor']} and Theorem \ref{['2-complex-correlation']}: $z_0=0$
  9. Proofs of Propositions \ref{['fTaylor']} and \ref{['gTaylor']}
  10. Proof of Theorem \ref{['2-complex-correlation']}: $z_0= 0$
  11. Properties on matrices and integrals

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 regular edge point that satisfies then as $N\to \infty$ the scaled correlation functions hold uniformly for all $\hat{z}_{1}, \ldots, \hat{z}_{n}$ in a compact subset of $\mathbb{C}$, where $\widetilde{r}_0= r_0 +R_0$ if $z_0=0,$ and $\widetilde{r}_0= r_0$ otherwise.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['RNn delta']}.
  • Proposition 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['2-complex-correlation']}: $z_0\neq 0$
  • proof : Proof of Proposition \ref{['fTaylor']}
  • ...and 5 more