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Analytic approach for the number statistics of non-Hermitian random matrices
- Source :
- Physical Review E. 103
- Publication Year :
- 2021
- Publisher :
- American Physical Society (APS), 2021.
-
Abstract
- We introduce a powerful analytic method to study the statistics of the number $\mathcal{N}_{\textbf{A}}(\gamma)$ of eigenvalues inside any contour $\gamma \in \mathbb{C}$ for infinitely large non-Hermitian random matrices ${\textbf A}$. Our generic approach can be applied to different random matrix ensembles, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable. The main outcome is an effective theory that determines the cumulant generating function of $\mathcal{N}_{\textbf{A}}$ via a path integral along $\gamma$, with the path probability distribution following from the solution of a self-consistent equation. We derive the expressions for the mean and the variance of $\mathcal{N}_{\textbf{A}}$ as well as for the rate function governing rare fluctuations of ${\mathcal{N}}_{\textbf{A}}{(\gamma)}$. All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.<br />Comment: 6 pages, 2 figures. SI as ancillary file
- Subjects :
- Physics
Random graph
FOS: Physical sciences
Disordered Systems and Neural Networks (cond-mat.dis-nn)
Condensed Matter - Disordered Systems and Neural Networks
Type (model theory)
01 natural sciences
Hermitian matrix
Jordan curve theorem
010305 fluids & plasmas
symbols.namesake
0103 physical sciences
Statistics
symbols
Probability distribution
010306 general physics
Rate function
Random matrix
Eigenvalues and eigenvectors
Subjects
Details
- ISSN :
- 24700053 and 24700045
- Volume :
- 103
- Database :
- OpenAIRE
- Journal :
- Physical Review E
- Accession number :
- edsair.doi.dedup.....10349675169e50b776649831ee15fe5f