We study the gap probabilities in the critical unitary invariant random matrix ensembles, where the Painleve II and Painleve XXXIV kernels arise. By studying the Fredholm determinants of the Painleve II and Painlev XXXIV kernels, we obtain integral expression of the gap probabilities by using solutions to the coupled Painleve II system. Moreover, the large gap asymptotics are derived with the constant terms given explicitly in terms of the Riemann zeta-function. This talk is based on a joint work with Dan Dai from City University of HongKong.
Prof. Xu Shuaixia got his PhD degree from Sun Yat-sen University in 2011. From 2011 to 2013, he worked as a postdoctoral fellow at Sun Yat-sen and is currently a lecturer at Sun Yat-sen University. His research areas include nuclear science and technology-other disciplines of nuclear science and technology.