Gessel-Type Expansion for the Circular ${\hat{I}}^2$-Ensemble and Central Limit Theorem for the Sine-${\hat{I}}^2$ Process for ${\hat{I}}^2\le 2$

Abstract

We obtain a Gessel-type expansion in Jack polynomials for the expectations of multiplicative functionals in the circular ${\hat{I}}^2$-ensemble. As a consequence, we establish a Szegő-type limit theorem for all $H^{1/2}(\mathbb{T})$ functions when ${\hat{I}}^2\le 2$, 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-${\hat{I}}^2$ process and yields a Soshnikov-type central limit theorem for the sine-${\hat{I}}^2$ process in the full $H^{1/2}(\mathbb{R})$ class.

25 pages; title changed, proof of Proposition 1.5 is simplified, subsection 1.3 is replaced by section 6