Gessel-Type Expansion for the Circular $β$-Ensemble and Central Limit Theorem for the Sine-$β$ Process for $β\le 2$

Abstract

We obtain a Gessel-type expansion in Jack polynomials for the expectations of multiplicative functionals in the circular $β$-ensemble. As a consequence, we establish a Szegő-type limit theorem for all $H^{1/2}(\mathbb{T})$ functions when $β\le 2$, together with an explicit rate of convergence for functions from $H^1(\mathbb{T})$. The estimate is stable under the scaling limit to the sine-$β$ process and yields a Soshnikov-type central limit theorem for the sine-$β$ process in the full $H^{1/2}(\mathbb{R})$ class.