PreprintCC BY 4.0arXiv
Averages of traces and ratios of characteristic polynomials of random unitary matrices
Abstract
In this work we explore the extent to which the Ratios Theorem for averages over U(N), with respect to Haar measure, of ratios of characteristic polynomials can be used to evaluate averages of traces of unitary matrices. We derive concrete expressions for all averages involving up to 4 traces, valid for any matrix size, N. We also calculate the so-called zero-swap and one-swap contributions to an average with any number of traces, which amounts to determining general trace moments in a range double the size of the renowned result of Diaconis and Shahshahani. The expression is tractable enough to use in applications, for example to determine the parameter range where certain trace averages assume negative values.
Related
- On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices
- Painlevé representations for joint moments of characteristic polynomials from the orthogonal and unitary symplectic groups
- Ehrhart polynomials of cyclic polytopes as averages of zonotope Ehrhart polynomials
- Push forwards of smooth measures on the unitary group in odd characteristic under the Chevalley map are almost bounded
- Counting Integral Matrices with Given Cubic Characteristic Polynomial
BibTeX
@misc{bendel2026averages,
title = {{Averages of traces and ratios of characteristic polynomials of random unitary matrices}},
author = {Ada Bendel and Rhys Cahill and John Brian Conrey and Gregor Harrison Barraclough and Ayesha Irfan and Nina Claire Snaith and Leonid Tyulenev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32236},
eprint = {2610.32236},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32236},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Averages of traces and ratios of characteristic polynomials of random unitary matrices
AU - Ada Bendel
AU - Rhys Cahill
AU - John Brian Conrey
AU - Gregor Harrison Barraclough
AU - Ayesha Irfan
AU - Nina Claire Snaith
AU - Leonid Tyulenev
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32236
UR - https://arxiv.org/abs/2610.32236
AB - In this work we explore the extent to which the Ratios Theorem for averages over U(N), with respect to Haar measure, of ratios of characteristic polynomials can be used to evaluate averages of traces of unitary matrices. We derive concrete expressions for all averages involving up to 4 traces, valid for any matrix size, N. We also calculate the so-called zero-swap and one-swap contributions to an average with any number of traces, which amounts to determining general trace moments in a range double the size of the renowned result of Diaconis and Shahshahani. The expression is tractable enough to use in applications, for example to determine the parameter range where certain trace averages assume negative values.
ER -