English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The circular $\beta$ ensemble for $\beta =1,2$ and 4 corresponds to circularorthogonal, unitary and symplectic ensemble respectively as introduced byDyson. The statistical state of the eigenvalues is then a determinantal pointprocess ($\beta = 2$) and Pfaffian point process ($\beta = 1,4$). The explicitfunctional 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 spacingdistributions and their generating function. We use $\sigma$-Painlev\'echaracterisations to show that the functional form of the first correction isrelated to the leading term via a second derivative. Explicit functional formsare used to show that the spectral form factors for $\beta =1,2$ and 4 alsoadmit an asymptotic expansion in $1/N^2$. Differential relations are identifiedexpressing the first and second correction in terms of the limiting functionalform, and evidence is presented that they hold for general $\beta$. For even$\beta$ it is proved that the two-point correlation function permits anasymptotic expansion in $1/N^2$, and moreover that the leading correctionrelates to the limiting functional form via a second derivative.

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