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

Abstract

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 Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $\alpha =\beta$). By exploiting the even parity of the weight, a doubling of the interval 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 the smallest 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 the log 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 the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes $G-$ function valuated at special point.

38 pages

References in corpus (4)

• Painleve IV and degenerate Gaussian Unitary Ensembles
• Orthogonal polynomials with discontinuous weights
• Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite $n$ to Double Scaling
• Continuous and discrete Painleve equations arising from the gap probability distribution of the finite $n$ Gaussian Unitary Ensembles

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• Symmetric Sextic Freud Weight
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• The Smallest Eigenvalue Distribution of the Jacobi Unitary Ensembles