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arXiv · 1209.2190

Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

N. S. Witte·F. Bornemann·P. J. Forrester
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Abstract

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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N. S. Witte, F. Bornemann, P. J. Forrester. 2012-09-11. Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles. https://doi.org/10.1088/0951-7715%2F26%2F6%2F1799

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