Free versions of the strong Szegő limit theorem

Michael T. Jury; Lodewyk J. van Rensburg; George Roman

Free versions of the strong Szegő limit theorem

Operator Algebras 2026-07-28 v1 Functional Analysis

Authors: Michael T. Jury , Lodewyk J. van Rensburg , George Roman

Abstract

The Strong Szeg\H{o} Limit Theorem is a theorem about the asymptotics of the determinants of large Toeplitz matrices. It can be reformulated as a probabilistic statement about eigenvalue statistics of random unitary matrices. We prove a multivariate generalization of the theorem in this latter form, replacing a single unitary with a system of independent random unitaries. It turns out that the standard proofs of the classical theorem do not generalize to this setting, and instead we must import tools from random matrix theory, free probability, and noncommutative function theory. In addition, we obtain results about the averaged determinants of random unitary pencils, as the size of the unitaries tends to infinity; and an auxiliary result giving a formula for the spectral radius of a matricial sum of free Haar unitaries.

Keywords

Toeplitz matrices random unitary matrices free probability noncommutative function theory Szegő limit theorem eigenvalue statistics

Cite

@article{arxiv.2607.25980,
  title  = {Free versions of the strong Szegő limit theorem},
  author = {Michael T. Jury and Lodewyk J. van Rensburg and George Roman},
  journal= {arXiv preprint arXiv:2607.25980},
  year   = {2026}
}