The limit empirical spectral distribution of complex matrix polynomials

Giovanni Barbarino*, Vanni Noferini

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


We study the empirical spectral distribution (ESD) for complex n × n matrix polynomials of degree k under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) n →∞ with k constant and (2) k →∞ with n bounded by O(kP) for some P > 0; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on k (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

Original languageEnglish
Article number2250023
JournalRandom Matrices: Theory and Application
Publication statusE-pub ahead of print - 2021
MoE publication typeA1 Journal article-refereed


  • Companion matrix
  • Empirical spectral distribution
  • Logarithmic potential
  • Polynomial eigenvalue problem
  • Random matrix polynomial
  • Universality


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