Abstract
We introduce a powerful analytic method to study the statistics of the number of eigenvalues inside any smooth Jordan curve for infinitely large non-Hermitian random matrices . Our generic approach can be applied to different random matrix ensembles of a mean-field type, 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, and obtain explicit results for the diluted real Ginibre ensemble. The main outcome is an effective theory that determines the cumulant generating function of via a path integral along , with the path probability distribution following from the numerical solution of a nonlinear self-consistent equation. We derive expressions for the mean and the variance of as well as for the rate function governing rare fluctuations of . All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.
- Received 28 July 2020
- Revised 15 April 2021
- Accepted 18 May 2021
DOI:https://doi.org/10.1103/PhysRevE.103.062108
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