The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

Giovanni Barbarino () and Vanni Noferini ()
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Giovanni Barbarino: Aalto University
Vanni Noferini: Aalto University

Journal of Theoretical Probability, 2023, vol. 36, issue 1, 99-133

Abstract: Abstract We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex $$n \times n$$ n × n Gaussian monic matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) $$n \rightarrow \infty $$ n → ∞ with k constant and (2) $$k \rightarrow \infty $$ k → ∞ with n constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: We slightly extend a result by Bürgisser and Cucker on the tail bound for the norm of the pseudoinverse of a nonzero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.

Keywords: Random matrix polynomial; Empirical spectral distribution; Polynomial eigenvalue problem; Strong circle law; Companion matrix; 60B20; 15B52; 15A15; 65F15; 65F20 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01163-3

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