Mesoscopic Edge Universality of Orthogonal Polynomial Ensembles

TL;DR

The paper addresses mesoscopic edge fluctuations of orthogonal polynomial ensembles (OPEs) with both continuous and discrete measures and proves a universal central limit theorem for linear statistics at scales $n^{\alpha}$. The authors develop a novel edge-specific resolvent analysis for truncated Jacobi matrices with slowly varying recurrence coefficients, overcoming the limitations of the Combes–Thomas estimate. They derive a universal variance formula $\sigma_f^2=\frac{1}{8\pi^2}\iint\left(\frac{f(x)-f(y)}{x-y}\right)^2dxdy$ adapted to edge geometry, and show the result holds for varying and non-varying weights, including classical ensembles (Jacobi, Laguerre, GUE) and discrete models from random tilings (Hahn, Krawtchouk, Tricomi–Carlitz). The analysis relies on cumulant expansions tied to Jacobi recurrence relations, a careful truncation scheme, and a strong Szegő-type determinant limit, yielding a robust universality mechanism at mesoscopic edges with broad applicability. This advances understanding of edge mesoscopic universality and provides tools applicable to tiling models and other extended-OPEs beyond bulk results.

Abstract

In this paper, we study the mesoscopic fluctuations at edges of orthogonal polynomial ensembles with both continuous and discrete measures. Our main result is a Central limit Theorem (CLT) for linear statistics at mesoscopic scales. We show that if the recurrence coefficients for the associated orthogonal polynomials are slowly varying, a universal CLT holds. Our primary tool is the resolvent for the truncated Jacobi matrices associated with the orthogonal polynomials. While the Combes-Thomas estimate has been successful in obtaining bulk mesoscopic fluctuations in the literature, it is too rough at the edges. Instead, we prove an estimate for the resolvent of Jacobi matrices with slowly varying entries. Particular examples to which our CLT applies are Jacobi, Laguerre and Gaussian unitary ensembles as well as discrete ensembles from random tilings.

Figures (6)

• Figure 1: The equilibrium measure of Laguerre Unitary Ensemble. Hard edge $x=0$, soft edge $x=4$.
• Figure 2: Illustration of Proposition \ref{['thm: overview F entry estimation']}. Given the entries of $J^{(r)}_{n+2mn^\beta}$ and $J^{(r)}_{n\pm2mn^\beta}$ with both indices ranging between $n-2mn^\beta$ and $n+2mn^\beta$ are well behaved, we have that the entries of $\left( J^{(r)}_{n+2mn^\beta} \right)^{-1}$ and $\left( J^{(r)}_{n\pm2mn^\beta} \right)^{-1}$ are of exponentially small in the shaded areas respectively. Note that the matrices are block matrices and we ignore the trivial blocks in the illustration.
• Figure 3: The equilibrium measure of Laguerre Unitary Ensemble.
• Figure 4: The equilibrium measure of Gaussian Unitary Ensemble.
• Figure 5: The equilibrium measure of Jacobi Unitary Ensemble.

Theorems & Definitions (74)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Corollary 3.1
• proof
• Corollary 3.2