酉系综软边第一和第二特征值的联合分布
Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles
摘要：
第一和第二特征值在酉系综的软边的联合分布的密度函数被发现在Painlevé II超越及其相关的isomonodromic系统。作为推论，这两个本征值之间的间距的密度函数具有类似的特征。由此产生的Painlevé II的特解是Hasting-McLeod解的双移位Bäcklund变换，它适用于最大本征值分布在软边的情况。我们的推论是通过采用硬到软的边缘过渡到现有的结果的联合分布的第一和第二个特征值在硬边缘\cite{FW_2007}。此外，递归下的$a \mapsto a+1$的数量指定后者获得。一个弗雷德霍姆行列式型特征是用来提供准确的数值之间的两个最大的特征值的间距的分布。
英文摘要：
The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlevé II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painlevé II that arises is a double shifted Bäcklund transformation of the Hasting-McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hard-to-soft edge transitions to existing results for the joint distribution of the first and second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under $a \mapsto a+1$ of quantities specifying the latter are obtained. A Fredholm determinant type characterisation is used to provide accurate numerics for the distribution of the spacing between the two largest eigenvalues.
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