Non universality of fluctuations of outliers for Hermitian polynomials in a complex Wigner matrix and a spiked diagonal matrix

TL;DR

This paper investigates the non-universal fluctuation behavior of outliers in Hermitian polynomials of complex Wigner matrices combined with spiked diagonal matrices, extending previous results to a broader class of polynomial deformations.

Contribution

It generalizes the non-universality of outlier fluctuations from additive deformations to any Hermitian polynomial in the setting of complex Wigner matrices.

Findings

Outliers exhibit non-universal fluctuations described by operator-valued subordination functions.

The results extend previous non-universality phenomena to all Hermitian polynomial deformations.

Theoretical framework uses free probability theory to characterize fluctuations.

Abstract

We study the fluctuations associated to the a.s. convergence, established by Belinschi-Bercovici-Capitaine, of the outliers of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the non universality phenomenon previously established for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The result is described using the operator-valued subordination functions of free probability theory.