PreprintCC BY 4.0arXiv
Gaussian limits for traces of circular beta-ensembles
Abstract
Consider the circular beta-ensembles, Jiang and Matsumoto computed the moments of traces, and proved central limit theorems for polynomial linear statistics using Jack functions. We give a direct proof of the Gaussian limits for the traces of powers of circular beta-ensembles using the theory of orthogonal polynomials on the unit circle and Stein's method. The proof relies on an explicit coupling of the Verblunsky coefficients of circular beta-ensembles, which also allow us to obtain almost sure and L2 convergence.
Related
- Bulk limits of the truncated circular beta ensembles
- Exploring continuous beta-ensembles: A Python implementation for random matrix spectral statistics
- Fisher--Hartwig asymptotics and critical moments for the circular beta ensemble
- A large deviation principle for the Gaussian beta ensemble in the high temperature regime
- Convergence of Kikuchi matrices to Γ-independent and q-Gaussian limits
BibTeX
@misc{li2026gaussian,
title = {{Gaussian limits for traces of circular beta-ensembles}},
author = {Yun Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01803},
eprint = {2610.01803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01803},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian limits for traces of circular beta-ensembles
AU - Yun Li
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01803
UR - https://arxiv.org/abs/2610.01803
AB - Consider the circular beta-ensembles, Jiang and Matsumoto computed the moments of traces, and proved central limit theorems for polynomial linear statistics using Jack functions. We give a direct proof of the Gaussian limits for the traces of powers of circular beta-ensembles using the theory of orthogonal polynomials on the unit circle and Stein's method. The proof relies on an explicit coupling of the Verblunsky coefficients of circular beta-ensembles, which also allow us to obtain almost sure and L^2 convergence.
ER -