Determinants of Random Unitary Pencils

TL;DR

This paper studies the determinants of random unitary pencils, providing exact formulas for scalar cases and conjectures for the general case, advancing understanding of their statistical properties.

Contribution

It introduces new formulas for determinants of random unitary pencils and proposes an asymptotic conjecture, with partial proof, expanding theoretical knowledge in random matrix theory.

Findings

Exact formula for scalar coefficient determinants

Conjectured asymptotic formula for general case

Proof of a special case of the conjecture

Abstract

We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by integrating against the Haar measure on the unitary group. We obtain an exact formula in the case of scalar coefficients, and conjecture an asymptotic formula in the general case, and prove a special case of the conjecture.