Gap probabilities in non-Hermitian random matrix theory
Gap probabilities in non-Hermitian random matrix theory
复制标题
非厄米随机矩阵理论中的间隙概率
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
L. Shifrin
中科院分区：
文献类型：
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作者：
G. Akemann;M. J. Phillips;L. Shifrin
我们计算半径为r的圆围绕原点恰好包含k个复本征值的间隙概率。我们考虑了四种不同的随机矩阵系综：吉尼伯系综及其手性络合物对应物，具有复数(β=2)或四元数实(β=4)矩阵元素。对于一般的非高斯权，我们分别给出了取决于非赫米性参数的Fredholm型行列式或Pfaffian表示。在最大非厄米性下，也就是对于旋转不变的权重，β=4的Fredholm本征值的乘积从β=2的情况出发，跳过每两个因子，与厄米特系综的已知关系相反。在选择高斯权的基础上，给出了手征情况下Fredholm本征值的Bessel-K和不完全Bessel-I函数的新的显式表达式。这与Ginibre系综在不完全指数方面的已知结果形成了对比。进一步，我们给出了对数的渐近展开式。
We compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (β=2) or quaternion real (β=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is, for rotationally invariant weights, the product of Fredholm eigenvalues for β=4 follows from the β=2 case by skipping every second factor, in contrast to the known relation for Hermitian ensembles. On additionally choosing Gaussian weights we give new explicit expressions for the Fredholm eigenvalues in the chiral case, in terms of Bessel-K and incomplete Bessel-I functions. This compares with known results for the Ginibre ensembles in terms of incomplete exponentials. Furthermore, we present an asymptotic expansion of the log...