The Singular Values of the GUE (Less is More)

TL;DR

This paper reveals a surprising decomposition of GUE singular values into two independent Laguerre ensembles, connecting classical laws and providing new insights into eigenvalue distributions, determinants, and condition numbers.

Contribution

It introduces a novel decomposition of GUE singular values into independent Laguerre ensembles, linking classical laws and offering new analytical tools.

Findings

Singular values of GUE are distributed as union of two independent Laguerre ensembles.

Connection between semicircle law and quarter-circle law via Hermite and Laguerre polynomials.

Absolute value of GUE determinant expressed as product of independent variables, revealing log-normality.

Abstract

Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large $n$ limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter $\pm$1/2. Similarly, we write the absolute value of the determinant of the $n\times{}n$ GUE as a product n independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.