一类高斯酉系综的间隙概率与PainlevéⅡ和Heun方程

The Gap Probability of a Deformed Gaussian Unitary Ensemble PainlevéⅡ and Heun Equations

本文主要讨论了一类扰动的高斯权函数|z|^(α)exp(-z^(2)),z∈R,α>0生成的酉系综在区间(-t,t),t>0上的间隙概率问题.通过正交于该权函数的多项式满足的一组阶梯算子,得到了决定这个间隙概率的两个参量函数R_(n)(t)和r_(n)(t),其中r_(n)(t)满足一个离散PainlevéⅡ方程.基于另一个参量函数R_(n)(t)在双尺度下的渐近行为,以及该类正交多项式满足的二阶常微分方程得到一类双合流Heun方程.

In this paper,we focus on the gap probability of the unitary ensemble generated by the deformed Gaussian weight|z|^(α)exp(-z^(2)),z∈R,α>0 on the interval(-t,t),t>0.By using the ladder operator method of orthogonal polynomials associated with the weight,two quantities which determines the gap probability,are denoted by R_(n)(t)and r_(n)(t).We find r_(n)(t)satisfies a discrete PainlevéⅡequation by algebraic transformation.We also show that the second order differential equation of orthogonal polynomials approximately satisfies a bi-confluent Heun equation basing on the asymptotic behavior of the other quantity R_(n)(t).