Spectral statistics of non-Hermitian random matrix ensembles

Recently Burkhardt et al. introduced the [Formula: see text]-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but [Formula: see text] of the eigenvalues are on the order of [Formula: see text] and converge to semi-circular behavior, with the remaining [Formula: see text] of size [Formula: see text] and converging to hollow Gaussian ensembles). We generalize their work to consider non-Hermitian ensembles with complex eigenvalues; instead of a blip new behavior is seen, ranging from multiple satellites to annular rings. These results are based on moment method techniques adapted to the complex plane as well as analysis of singular values.

Download Full-text

Filling up complex spectral regions through non-Hermitian disordered chains

Chinese Physics B ◽

10.1088/1674-1056/ac4a73 ◽

2022 ◽

Author(s):

Hui Jiang ◽

Ching Hua Lee

Keyword(s):

Complex Plane ◽

Random Matrix Theory ◽

Random Matrix ◽

Matrix Theory ◽

Quantum Circuits ◽

High Tolerance ◽

Stochastic Nature ◽

Random Matrix Ensembles ◽

Matrix Ensembles ◽

Very High

Abstract Eigenspectra that fill regions in the complex plane have been intriguing to many, inspiring research from random matrix theory to esoteric semi-infinite bounded non-Hermitian lattices. In this work, we propose a simple and robust ansatz for constructing models whose eigenspectra fill up generic prescribed regions. Our approach utilizes specially designed non-Hermitian random couplings that allow the co-existence of eigenstates with a continuum of localization lengths, mathematically emulating the effects of semi-infinite boundaries. While some of these couplings are necessarily long-ranged, they are still far more local than what is possible with known random matrix ensembles. Our ansatz can be feasibly implemented in physical platforms such as classical and quantum circuits, and harbors very high tolerance to imperfections due to its stochastic nature.

Download Full-text

Non-Hermitian ensembles

10.1093/oxfordhb/9780198744191.013.18 ◽

2018 ◽

Author(s):

Alexei Morozov

Keyword(s):

Random Matrix ◽

Matrix Theory ◽

Ginibre Ensemble ◽

Fold Family ◽

Large Matrix ◽

Complex Eigenvalues ◽

Wide Range ◽

Random Matrix Ensembles ◽

Matrix Ensembles ◽

Physical Phenomena

This article discusses the three-fold family of Ginibre random matrix ensembles (complex, real, and quaternion real) and their elliptic deformations. It also considers eigenvalue correlations that are exactly reduced to two-point kernels in the strongly and weakly non-Hermitian limits of large matrix size. Ginibre introduced the complex, real, and quaternion real random matrix ensembles as a mathematical extension of Hermitian random matrix theory. Statistics of complex eigenvalues are now used in modelling a wide range of physical phenomena. After providing an overview of the complex Ginibre ensemble, the article describes random contractions and the complex elliptic ensemble. It then examines real and quaternion-real Ginibre ensembles, along with real and quaternion-real elliptic ensembles. In particular, it analyses the kernel in the elliptic case as well as the limits of strong and weak non-Hermiticity.

Download Full-text

Random matrix ensembles with split limiting behavior

Random Matrices Theory and Application ◽

10.1142/s2010326318500065 ◽

2018 ◽

Vol 07 (03) ◽

pp. 1850006

Author(s):

Paula Burkhardt ◽

Peter Cohen ◽

Jonathan DeWitt ◽

Max Hlavacek ◽

Steven J. Miller ◽

...

Keyword(s):

Random Matrix ◽

Spectral Measure ◽

Symmetric Matrix ◽

Weight Functions ◽

Random Real ◽

Limiting Behavior ◽

New Family ◽

Random Matrix Ensembles ◽

Matrix Ensembles ◽

Limiting Spectral Measure

We introduce a new family of [Formula: see text] random real symmetric matrix ensembles, the [Formula: see text]-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but [Formula: see text] eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as [Formula: see text]; the remaining [Formula: see text] are tightly constrained near [Formula: see text] and their distribution converges to the [Formula: see text] hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for complex and quaternionic analogues. We are able to isolate each regime separately through appropriate choices of weight functions for the eigenvalues and then an analysis of the resulting combinatorics.

Download Full-text