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<p>We study mesoscopic fluctuations of orthogonal polynomial ensembles on the unit circle. We show that asymptotics of such fluctuations are stable under decaying perturbations of the recurrence coefficients, where the appropriate decay rate depends on the scale considered. By directly proving Gaussian limits for certain constant coefficient ensembles, we obtain mesoscopic scale Gaussian limits for a large class of orthogonal polynomial ensembles on the unit circle. As a corollary we prove mesoscopic central limit theorems (for all mesoscopic scales) for the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\beta =2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> circular Jacobi ensembles with real parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\delta &gt;-1/2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>&gt;</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Mesoscopic Universality for Circular Orthogonal Polynomial Ensembles

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月3日

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  • Jonathan Breuer,
  • Daniel Ofner

摘要

We study mesoscopic fluctuations of orthogonal polynomial ensembles on the unit circle. We show that asymptotics of such fluctuations are stable under decaying perturbations of the recurrence coefficients, where the appropriate decay rate depends on the scale considered. By directly proving Gaussian limits for certain constant coefficient ensembles, we obtain mesoscopic scale Gaussian limits for a large class of orthogonal polynomial ensembles on the unit circle. As a corollary we prove mesoscopic central limit theorems (for all mesoscopic scales) for the \(\beta =2\) β = 2 circular Jacobi ensembles with real parameter \(\delta >-1/2\) δ > - 1 / 2 .