Gap Probability Distribution of Gaussian Unitary Ensembles and Painlevé V Equation

Shengjie Zhang, Shulin Lyu

Published 2024-10-22Version 1

We consider the Hankel determinant generated by the moments of the even weight function ${\rm e}^{-x^2}(A+B\theta(x^2-a^2)), x\in(-\infty,+\infty), a>0, A\ge0, A+B\ge0$. It is intimately related to the gap probability of the Gaussian unitary ensembles on $(-a,a)$ or $(-\infty,-a)\cup(a,+\infty)$. We derive the ladder operators for the monic polynomials orthogonal with respect to this weight function and three supplementary conditions. By using them and differentiating the orthogonality relation, we get difference and Riccati equations for the two auxiliary quantities introduced in the ladder operators. From these equations, we obtain a second-order difference equation and a second-order second-degree ordinary differential equation satisfied by the coefficient of the three-term recurrence relation for the monic orthogonal polynomials. Moreover, we establish the second-order fourth-degree ordinary differential equation satisfied by the logarithmic derivative of the Hankel determinant. As $n\to\infty$ and $a\to0^{+}$ such that $\tau=2\sqrt{2n}a$ is fixed, this equation is reduced to the $\sigma$-form of a Painlev\'{e} V equation in the variable $\tau$.