Determinants of Random Unitary Pencils

2025/06/04 by Jury, Michael T., Roman, George
#15A15 #15A52 #46E22 #46L54 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2506.04400

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.

Citations

• Factoring determinants and applications to number theory
• An invitation to the Drury-Arveson space
• Non-commutative rational functions in the full Fock space
• The outer spectral radius and dynamics of completely positive maps
• On domains of noncommutative rational functions
• Limiting Spectral Distribution of Sum of Unitary and Orthogonal Matrices
• Averages of Characteristic Polynomials in Random Matrix Theory
• Combinatorial aspects of Connes's embedding conjecture and asymptotic distribution of traces of products of unitaries
• Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability
• Algebraic aspects of increasing subsequences
• The invariant theory of n × n matrices

Related

• A new approach to the characteristic polynomial of a random unitary matrix
• Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts
• Moments of Moments of the Characteristic Polynomials of Random Orthogonal and Symplectic Matrices
• Powers of large random unitary matrices and Toeplitz determinants
• Moments of the characteristic polynomial in the three ensembles of random matrices