### Abstract

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples - adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erdős-Rényi graphs, and adjacency matrices of weighted oriented Erdős-Rényi graphs - we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.

Original language | English |
---|---|

Article number | 434003 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 52 |

Issue number | 43 |

Early online date | 26 Apr 2019 |

DOIs | |

Publication status | Published - 1 Oct 2019 |

### Keywords

- cond-mat.stat-mech
- cond-mat.dis-nn
- math-ph
- math.MP
- non-Hermitian matrices
- random matrix theory
- complex networks
- sparse matrices

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Statistics and Probability
- Mathematical Physics
- Modelling and Simulation

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*52*(43), [434003]. https://doi.org/10.1088/1751-8121/ab1ce0

**Spectral Theory of Sparse Non-Hermitian Random Matrices.** / Metz, Fernando Lucas; Neri, Izaak; Rogers, Tim.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 52, no. 43, 434003. https://doi.org/10.1088/1751-8121/ab1ce0

}

TY - JOUR

T1 - Spectral Theory of Sparse Non-Hermitian Random Matrices

AU - Metz, Fernando Lucas

AU - Neri, Izaak

AU - Rogers, Tim

N1 - Invited Topical Review

PY - 2019/10/1

Y1 - 2019/10/1

N2 - Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples - adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erdős-Rényi graphs, and adjacency matrices of weighted oriented Erdős-Rényi graphs - we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.

AB - Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples - adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erdős-Rényi graphs, and adjacency matrices of weighted oriented Erdős-Rényi graphs - we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.

KW - cond-mat.stat-mech

KW - cond-mat.dis-nn

KW - math-ph

KW - math.MP

KW - non-Hermitian matrices

KW - random matrix theory

KW - complex networks

KW - sparse matrices

UR - http://www.scopus.com/inward/record.url?scp=85069812488&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/ab1ce0

DO - 10.1088/1751-8121/ab1ce0

M3 - Article

VL - 52

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 43

M1 - 434003

ER -