Limiting Spectral Radii for Products of Ginibre Matrices and Their Inverses

Xiansi Ma and Yongcheng Qi ()
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Xiansi Ma: University of Minnesota Duluth
Yongcheng Qi: University of Minnesota Duluth

Journal of Theoretical Probability, 2024, vol. 37, issue 4, 3756-3780

Abstract: 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 .

Keywords: Product matrix; Eigenvalue; Spectral radius; Ginibre matrix; 60B20; 60F05; 60G70 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-024-01341-5

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