# Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles

Lyu, Shulin (School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong, 519082, China) ; Chen, Yang (Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China) ; Fan, Engui (School of Mathematical Sciences, Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai, 200433, China)

04 December 2017

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 α=β ). By exploiting the even parity of the weight, a doubling of the interval to (a2,∞) 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 the LUE with parameter α=−1/2 , and α=1/2 and the (shifted) JUE with weights x1/2(1−x)β and x−1/2(1−x)β .

Published in: Nuclear Physics B (2017)