Gaussian limits for traces of circular beta-ensembles

Yun Li

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arXiv · 2610.01803

Gaussian limits for traces of circular beta-ensembles

Yun Li

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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 $L^2$ convergence.

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Yun Li. 2026-10-01. Gaussian limits for traces of circular beta-ensembles. https://arxiv.org/abs/2610.01803

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