Transformations of polynomial ensembles
Transformations of polynomial ensembles
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DOI：
10.1090/conm/661/13286
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发表时间：
2015-01
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通讯作者：
A. Kuijlaars
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文献类型：
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作者：
A. Kuijlaars
多项式系综是关于$n$实粒子的位置的概率密度函数，对于某些函数$f_1，\ldots，f_n$，形式为$\frac{1}{Z_n}\，\prod_{j<；k}(x_k-x_j)\，\det\Left[f_k(X_J)\right]_{j，k=1}^n$。这样的系综经常出现为随机矩阵的联合特征值密度。我们给出了一些保持多项式系综结构的变换。这些变换包括通过删除一行和一列来限制厄米矩阵，对厄米矩阵进行一阶修改，以及通过增加具有复高斯的额外行和列来扩展厄米矩阵。
A polynomial ensemble is a probability density function for the position of $n$ real particles of the form $\frac{1}{Z_n} \, \prod_{j<k} (x_k-x_j) \, \det \left[ f_k (x_j) \right]_{j,k=1}^n$, for certain functions $f_1, \ldots, f_n$. Such ensembles appear frequently as the joint eigenvalue density of random matrices. We present a number of transformations that preserve the structure of a polynomial ensemble. These transformations include the restriction of a Hermitian matrix by removing one row and one column, a rank-one modification of a Hermitian matrix, and the extension of a Hermitian matrix by adding an extra row and column with complex Gaussians.