Conditional Haar measures on classical compact groups

TL;DR

This paper introduces a probabilistic approach to Haar measures on classical compact groups, deriving new limit theorems for derivatives of characteristic polynomials conditioned on stable subspaces.

Contribution

It provides a novel probabilistic proof of the Weyl integration formula and establishes explicit distributions and limit theorems for conditioned Haar measures on classical groups.

Findings

Distribution of derivatives of characteristic polynomials derived

Central limit theorem for log of derivatives established

Asymptotic behavior of density near zero analyzed

Abstract

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension $n$. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension $p$. The developed method leads to the following result: for this conditional measure, writing $Z_U^{(p)}$ for the first nonzero derivative of the characteristic polynomial at 1, \[\frac{Z_U^{(p)}}{p!}\stackrel{\mathrm{law}}{=}\prod_{\ell =1}^{n-p}(1-X_{\ell}),\] the $X_{\ell}$'s being explicit independent random variables. This implies a central limit theorem for $\log Z_U^{(p)}$ and asymptotics for the density of $Z_U^{(p)}$ near 0. Similar limit theorems are given for the orthogonal and symplectic groups, relying on results of Killip and Nenciu.