On the rightmost eigenvalue of non-Hermitian random matrices. (English) Zbl 1536.15036
Consider a square complex matrix \(X\) with independent and identically distributed random coefficients \(x_{ij}, 1 \leq i, j \leq n\), with \[x_{ij} \overset{d}{=} n^{-1/2}\chi,\] where \(\chi\) is a centered complex random variable with \(\mathbb{E} |\chi|^2 = 1\), \(\mathbb{E} \chi^2 = 0\) and bounded higher moments.
The authors give asymptotics for the real part of the rightmost eigenvalue of \(X\). Asymptotics of the same form were obtained by the authors for the Ginibre ensemble in [J. Math. Phys. 63, No. 10, Article ID 103303, 11 p. (2022; Zbl 1507.15023)]. In fact, this result shows that the three terms asymptotics they obtain is universal. Understanding the real part of the rightmost eigenvalue is of importance in bounding the growth rate of solutions of random linear systems of differential equations.
The result relies on the previous work by the authors [loc. cit.] for the Ginibre ensemble. The asymptotics of the Ginibre ensemble and of the i.i.d. model considered here are shown to be the same using a Green function comparison theorem, whose use relies on the Hermitization theorem by V. L. Girko [Teor. Veroyatn. Primen. 29, No. 4, 669–679 (1984; Zbl 0565.60034)].
The authors give asymptotics for the real part of the rightmost eigenvalue of \(X\). Asymptotics of the same form were obtained by the authors for the Ginibre ensemble in [J. Math. Phys. 63, No. 10, Article ID 103303, 11 p. (2022; Zbl 1507.15023)]. In fact, this result shows that the three terms asymptotics they obtain is universal. Understanding the real part of the rightmost eigenvalue is of importance in bounding the growth rate of solutions of random linear systems of differential equations.
The result relies on the previous work by the authors [loc. cit.] for the Ginibre ensemble. The asymptotics of the Ginibre ensemble and of the i.i.d. model considered here are shown to be the same using a Green function comparison theorem, whose use relies on the Hermitization theorem by V. L. Girko [Teor. Veroyatn. Primen. 29, No. 4, 669–679 (1984; Zbl 0565.60034)].
Reviewer: Thomas Buc-d’Alché (Lyon)
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 60B20 | Random matrices (probabilistic aspects) |
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