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/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

In this paper, we address a class of problems in unitary ensembles.Specifically, we study the probability that a gap symmetric about 0, i.e.$(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobiunitary ensembles (JUE) (where in the JUE, we take the parameters$\alpha=\beta$). By exploiting the even parity of the weight, a doubling of theinterval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE,shows that the gap probabilities maybe determined as the product of thesmallest eigenvalue distributions of the LUE with parameter $\alpha=-1/2,$ and$\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\beta}$ and$x^{-1/2}(1-x)^{\beta}$ The $\sigma$ function, namely, the derivative of thelog of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE,satisfies the Jimbo-Miwa-Okamoto $\sigma$ form of $P_{V}$ and $P_{VI}$,although in the shift Jacobi case, with the weight $x^{\alpha}(1-x)^{\beta},$the $\beta$ parameter does not show up in the equation. We also obtain theasymptotic expansions for the smallest eigenvalue distributions of the Laguerreunitary and Jacobi unitary ensembles after appropriate double scalings, andobtained the constants in the asymptotic expansion of the gap probablities,expressed in term of the Barnes $G-$ function valuated at special point.

Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles