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Limiting Spectral Radii for Products of Ginibre Matrices and Their Inverses

Author

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  • Xiansi Ma

    (University of Minnesota Duluth)

  • Yongcheng Qi

    (University of Minnesota Duluth)

Abstract

Consider the product of m independent n-by-n Ginibre matrices and their inverses, where $$m=p+q$$ m = p + q , p is the number of Ginibre matrices, and q is the number of inverses of Ginibre matrices. The maximum absolute value of the eigenvalues of the product matrices is known as the spectral radius. In this paper, we explore the limiting spectral radii of the product matrices as n tends to infinity and m varies with n. Specifically, when $$q\ge 1$$ q ≥ 1 is a fixed integer, we demonstrate that the limiting spectral radii display a transition phenomenon when the limit of p/n changes from zero to infinity. When $$q=0$$ q = 0 , the limiting spectral radii for Ginibre matrices have been obtained by Jiang and Qi [J Theor Probab 30: 326–364, 2017]. When q diverges to infinity as n approaches infinity, we prove that the logarithmic spectral radii exhibit a normal limit, which reduces to the limiting distribution for spectral radii for the spherical ensemble obtained by Chang et al. [J Math Anal Appl 461: 1165–1176, 2018] when $$p=q$$ p = q .

Suggested Citation

  • Xiansi Ma & Yongcheng Qi, 2024. "Limiting Spectral Radii for Products of Ginibre Matrices and Their Inverses," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3756-3780, November.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01341-5
    DOI: 10.1007/s10959-024-01341-5
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    References listed on IDEAS

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    1. Yongcheng Qi & Mengzi Xie, 2020. "Spectral Radii of Products of Random Rectangular Matrices," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2185-2212, December.
    2. Tiefeng Jiang & Yongcheng Qi, 2017. "Spectral Radii of Large Non-Hermitian Random Matrices," Journal of Theoretical Probability, Springer, vol. 30(1), pages 326-364, March.
    3. Chang, Shuhua & Qi, Yongcheng, 2017. "Empirical distribution of scaled eigenvalues for product of matrices from the spherical ensemble," Statistics & Probability Letters, Elsevier, vol. 128(C), pages 8-13.
    4. Tiefeng Jiang & Yongcheng Qi, 2019. "Empirical Distributions of Eigenvalues of Product Ensembles," Journal of Theoretical Probability, Springer, vol. 32(1), pages 353-394, March.
    Full references (including those not matched with items on IDEAS)

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