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The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

Author

Listed:
  • Giovanni Barbarino

    (Aalto University)

  • Vanni Noferini

    (Aalto University)

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.

Suggested Citation

  • Giovanni Barbarino & Vanni Noferini, 2023. "The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials," Journal of Theoretical Probability, Springer, vol. 36(1), pages 99-133, March.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-022-01163-3
    DOI: 10.1007/s10959-022-01163-3
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