[Paper Review] Small gaps of circular $\beta$-ensemble

Renjie Feng, Dongyi Wei|arXiv (Cornell University)|Jun 5, 2018

Random Matrices and Applications8 references3 citations

TL;DR

This paper establishes the limiting distribution of the smallest gaps in the circular β-ensemble (CβE) for any positive integer β. After normalization by $ n^{(eta+2)/(eta+1)} $, the gaps converge in distribution to a Poisson point process with an explicit intensity, yielding a limiting density for the k-th smallest gap proportional to $ x^{k(\beta+1)-1} e^{-x^{\beta+1}} $. The proof relies on novel asymptotic estimates of Selberg integrals and correlation functions.

ABSTRACT

In this article, we study the smallest gaps of the log-gas $\\beta$-ensemble on the unit circle (C$\\beta$E), where $\\beta$ is any positive integer. The main result is that the smallest gaps, after being normalized by $n^{\\frac {\\beta+2}{\\beta+1}}$, will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the $k$-th smallest gap, which is proportional to $x^{k(\\beta+1)-1}e^{-x^{\\beta+1}}$. In particular, the result applies to the classical COE, CUE and CSE in random matrix theory. The essential part of the proof is to derive several identities and inequalities regarding the Selberg integral, which should have their own interest.

Motivation & Objective

• To determine the limiting distribution of the smallest gaps in the circular β-ensemble (CβE) for general positive integer β.
• To extend previous results on CUE (β=2) and COE/CSE (β=1,4) to arbitrary β.
• To establish the universality of small gap statistics in random matrix theory beyond determinantal or Pfaffian processes.
• To derive precise asymptotic identities and inequalities for Selberg integrals that govern the correlation functions of CβE.

Proposed method

• Derive asymptotic estimates for partition functions of two-component log-gases with charges q=1 and q=k, using Selberg integral identities.
• Establish key inequalities and limits (e.g., Lemma 1.1 and Lemma 1.4) for $ C_{\beta,n-k,(k)} $ and related partition functions.
• Use the normalized gap process $ \chi^{(n)} $ with scaling $ \gamma = \frac{\beta+2}{\beta+1} $ to analyze local statistics.
• Apply moment convergence and Hölder’s inequality to control factorial moments of gap counts.
• Prove convergence in distribution to a Poisson point process by showing factorial moment convergence to Poisson limits.
• Use the intensity $ \lambda = \left( \frac{|I|A_\beta}{2\pi} \right)^k $ derived from asymptotic partition function ratios.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the smallest gaps in the circular β-ensemble for arbitrary positive integer β?
• RQ2 How does the scaling of the smallest gaps depend on β, and what normalization leads to a non-degenerate limit?
• RQ3 Can the Poisson limit for small gaps in CUE (β=2) be generalized to all β, and what is the explicit intensity of the limiting Poisson process?
• RQ4 What role do Selberg integral identities and their asymptotics play in deriving the joint correlation functions of CβE?
• RQ5 Is the small gap behavior universal across different β-ensembles, including COE (β=1), CUE (β=2), and CSE (β=4)?
• RQ6 How do the k-th smallest gap densities scale asymptotically, and what is their functional form?

Key findings

• The smallest gaps in the circular β-ensemble, when scaled by $ n^{(\beta+2)/(\beta+1)} $, converge in distribution to a Poisson point process with intensity $ \lambda = \left( \frac{|I|A_\beta}{2\pi} \right)^k $ for the k-th gap.
• The limiting density of the k-th smallest gap is proportional to $ x^{k(\beta+1)-1} e^{-x^{\beta+1}} $, explicitly derived from the Poisson limit.
• For β=2 (CUE), the result recovers the known limiting density proportional to $ x^{3k-1} e^{-x^3} $, confirming consistency with prior work.
• The asymptotic ratio $ \frac{C_{\beta,n-k,(k)}}{C_{\beta,n} n^{k(k-1)\beta/2}} \to A_{\beta,k} $ is established, with $ A_{\beta,k} $ given in closed form via gamma functions.
• The proof introduces new inequalities and asymptotic identities for Selberg integrals, particularly for two-component log-gases with mixed charges.
• The method applies beyond CβE: the results are extended to GOE in subsequent work and shown to be universal for Wigner matrices in later studies.