Finite size corrections in the bulk for circular $\beta$ ensembles

TL;DR

This paper investigates finite size corrections in the bulk for circular $eta$ ensembles, deriving asymptotic expansions and relations for correlation functions and spectral form factors, with implications for understanding eigenvalue distributions.

Contribution

It provides explicit asymptotic expansions in $1/N^2$ for correlation functions and spectral form factors in circular $eta$ ensembles, and relates corrections to the leading term via second derivatives.

Findings

Asymptotic expansion in $1/N^2$ for $n$-point correlation functions.

First correction related to the leading term via a second derivative.

Spectral form factors also admit an asymptotic expansion in $1/N^2$.

Abstract

The circular $\beta$ ensemble for $\beta =1,2$ and 4 corresponds to circular orthogonal, unitary and symplectic ensemble respectively as introduced by Dyson. The statistical state of the eigenvalues is then a determinantal point process ($\beta = 2$) and Pfaffian point process ($\beta = 1,4$). The explicit functional forms of the correlation kernels then imply that the general $n$-point correlation functions exhibit an asymptotic expansion in $1/N^2$, which moreover can be lifted to an asymptotic in $1/N^2$ for the spacing distributions and their generating function. We use $\sigma$-Painlev\'e characterisations to show that the functional form of the first correction is related to the leading term via a second derivative. In the case $\beta = 2$ this finding has immediate consequence in interpreting the empirical Riemann zeros spacing distribution at large height, and that of their thinning. Explicit functional forms are used to show that the spectral form factors for $\beta =1,2$ and 4 also admit an asymptotic expansion in $1/N^2$. Differential relations are identified expressing the first and second correction in terms of the limiting functional form, and evidence is presented that they hold for general $\beta$. For even $\beta$ it is proved that the two-point correlation function permits an asymptotic expansion in $1/N^2$, and moreover that the leading correction relates to the limiting functional form via a second derivative.