Zeros Of Random Analytic Functions And Spectral Properties Of Perturbed Unitary Matrices

2025/09/03 by Fares, Aniss
Computer Science · Mathematics · #15B52 (Primary) 47B93 (Secondary) #30B20 #60B20 #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2509.03252

openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the spectral properties of a rank-one multiplicative perturbation of a unitary matrix, a model introduced by Fyodorov. Building upon earlier results by Forrester and Ipsen, we provide a direct proof that the eigenvalues converge to the zeros of a specific Gaussian analytic function. Our approach extends these results to other unitarily invariant models. This method enables us to address a question raised by Dubach and Reker concerning the critical timescale at which an outlier emerges.

Citations

• Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series
• Dynamics of a rank-one multiplicative perturbation of a unitary matrix
• A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices
• From random matrices to random analytic functions

Related

• Dynamics of a rank-one multiplicative perturbation of a unitary matrix
• Rank 1 perturbations in random matrix theory -- a review of exact results
• Fluctuations for analytic test functions in the Single Ring Theorem
• Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes
• A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices