abstract: In this paper, we address a class of problems in unitary ensebles. 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 (a2,infty) for the GUE, and (a2,1), for the (symmetric) JUE shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of of the LUE with parameter alpha=-12, alpha=12 and the (shifted) JUE with weights x(12) (1-x)beta and x(-12) (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, satsifies the Jimbo-Miwa-Oakamoto sigma form for PV and PVI, although in the shifted 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 LUE, and the JUE after appropriate double scalings, and ontained the constants in the asymptotic expansion of the gap probabilities, expressed in term of the Barnes G-function evalauted at special point.
Yang CHEN (Macau), Shulin Lyu (Guangzhou) , and Engui FAN (Shanghai)