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Central limit theorem for the sine-$β$ point process at $β\le 2$

Sergei M. Gorbunov

TL;DR

The paper proves a Soshnikov-type central limit theorem for the sine-$β$ process in the regime $β\le 2$, by analyzing regularized additive functionals $\overline{S}_f$ with $f$ in the $1/2$-Sobolev class and studying their scaling limit under $f(x/R)$. The authors bridge circular $β$-ensembles and the sine process through Jack polynomial techniques, obtaining a Gessel-type expansion for multiplicative functionals via Jack measures and establishing sharp Laplace-transform bounds that yield Gaussian fluctuations and subgaussian tails. They prove convergence of additive functionals for the circular ensemble with $n$ particles to a Gaussian distribution for all $1/2$-Sobolev regular $f$ when $β\le 2$, and they derive a rate of convergence in Kolmogorov–Smirnov distance of order $(\ln R)^{-1/2}$. A key component is linking expectations of multiplicative functionals to Jack measures, generalizing the Schur-measure/Gessel connection to the Jack setting, and enabling a cohesive framework to translate from finite-$n$ ensembles to the sine limit. The results advance the understanding of fluctuations in $β$-ensembles, provide explicit quantitative bounds, and connect random matrix-type limits with symmetric-function theory through Jack polynomials and measures.

Abstract

The purpose of this paper is to establish the analogue of the Soshnikov Central Limit Theorem for the sine-$β$ process at $β\le 2$. We consider regularized additive functionals, which correspond to 1-Sobolev regular functions $f(x/R)$, in the limit $R\to\infty$. Their convergence to the Gaussian distribution with respect to the Kolmogorov-Smirnov metric at the rate $(\ln R)^{-1/2}$ is established. The proof is based on the convergence of the circular $β$-ensemble to the sine-$β$ process, which was shown by Killip and Stoiciu. We find a convenient bound for the Laplace transforms of additive functionals under the circular $β$-ensemble, which holds under the scaling limit, suggested by Killip and Stoiciu. Further, we show that the additive functionals under the circular $β$-ensemble with $n$ particles converge to the gaussian distribution as $n\to\infty$ for all $1/2$-Sobolev regular functions for $β\le 2$, as was conjectured by Lambert. Finally, in order to prove the limit theorem for the circular $β$-ensemble we derive the connection between expectations of multiplicative functionals and the Jack measures, which generalizes the connection between the circular unitary ensemble and the Schur measures given by Gessel's theorem.

Central limit theorem for the sine-$β$ point process at $β\le 2$

TL;DR

The paper proves a Soshnikov-type central limit theorem for the sine- process in the regime , by analyzing regularized additive functionals with in the -Sobolev class and studying their scaling limit under . The authors bridge circular -ensembles and the sine process through Jack polynomial techniques, obtaining a Gessel-type expansion for multiplicative functionals via Jack measures and establishing sharp Laplace-transform bounds that yield Gaussian fluctuations and subgaussian tails. They prove convergence of additive functionals for the circular ensemble with particles to a Gaussian distribution for all -Sobolev regular when , and they derive a rate of convergence in Kolmogorov–Smirnov distance of order . A key component is linking expectations of multiplicative functionals to Jack measures, generalizing the Schur-measure/Gessel connection to the Jack setting, and enabling a cohesive framework to translate from finite- ensembles to the sine limit. The results advance the understanding of fluctuations in -ensembles, provide explicit quantitative bounds, and connect random matrix-type limits with symmetric-function theory through Jack polynomials and measures.

Abstract

The purpose of this paper is to establish the analogue of the Soshnikov Central Limit Theorem for the sine- process at . We consider regularized additive functionals, which correspond to 1-Sobolev regular functions , in the limit . Their convergence to the Gaussian distribution with respect to the Kolmogorov-Smirnov metric at the rate is established. The proof is based on the convergence of the circular -ensemble to the sine- process, which was shown by Killip and Stoiciu. We find a convenient bound for the Laplace transforms of additive functionals under the circular -ensemble, which holds under the scaling limit, suggested by Killip and Stoiciu. Further, we show that the additive functionals under the circular -ensemble with particles converge to the gaussian distribution as for all -Sobolev regular functions for , as was conjectured by Lambert. Finally, in order to prove the limit theorem for the circular -ensemble we derive the connection between expectations of multiplicative functionals and the Jack measures, which generalizes the connection between the circular unitary ensemble and the Schur measures given by Gessel's theorem.
Paper Structure (21 sections, 31 theorems, 131 equations)

This paper contains 21 sections, 31 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. The circular $\beta$-ensemble
  3. The sine-$\beta$ process
  4. About the Jack measures
  5. Structure of the paper
  6. Acknowledgements
  7. Proof of Theorem \ref{['intro:gessel']}, Corollary \ref{['res:cbe_th']} and Theorem \ref{['res:cbe_est']}
  8. Jack polynomials
  9. Proof of Theorem \ref{['intro:gessel']} and Corollary \ref{['res:cbe_th']}
  10. Proof of Theorem \ref{['res:cbe_est']}
  11. Proof of Theorem \ref{['res:sb_th']}
  12. Regularization of additive functionals
  13. Proof of Proposition \ref{['outline:convergence']}
  14. Construction of point processes via random monotone increasing functions
  15. Convergence of point processes by linear interpolation
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $f$ be a continuous function on the unit circle. Then we have the expansion where $s_\lambda\mapsto s_\lambda(\rho_\pm)$ are homomorphisms from the algebra of symmetric functions $\Lambda\to \mathbb{C}$ defined on the Newton power sums by the formula

Theorems & Definitions (60)

  • Theorem 1.1: Gessel G_90TW_01
  • Theorem 1.2: Gessel-type expansion for the circular $\beta$-ensemble
  • Remark
  • Corollary 1.3: Central Limit Theorem for the circular-$\beta$ ensemble
  • Theorem 1.4
  • Proposition 1.5
  • Remark
  • Theorem 1.6: Central Limit Theorem for the sine-$\beta$ process
  • Remark
  • Corollary 1.7
  • ...and 50 more