We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex [Formula: see text] Ginibre ensemble, conditional on an arbitrary number [Formula: see text] of complex eigenvalues. These objects provide the simplest generalization of the expectations of the diagonal overlap ([Formula: see text]) and the off-diagonal overlap ([Formula: see text]) considered originally by Chalker and Mehlig. They also appear naturally in the problem of joint evolution of eigenvectors and eigenvalues for Brownian motions with values in complex matrices studied by the Krakow school. We find that these expectations possess a determinantal structure, where the relevant kernels can be expressed in terms of certain orthogonal polynomials in the complex plane. Moreover, the kernels admit a rather tractable expression for all [Formula: see text]. This result enables a fairly straightforward calculation of the conditional expectation of the overlap matrix in the local bulk and edge scaling limits as well as the proof of the exact algebraic decay and asymptotic factorization of these expectations in the bulk.

中文翻译：

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复 Ginibre 系综条件重叠的行列式结构

我们继续研究在 Chalker 和 Mehlig 的开创性论文中发起的特征向量和特征值的联合统计。我们研究的主要目标是对复数 [公式：见文本] Ginibre 集合的左右特征向量之间的重叠矩阵的期望，条件是复特征值的任意数量 [公式：见文本]。这些对象提供了对角重叠（[公式：参见文本]）和非对角重叠（[公式：参见文本]）最初由 Chalker 和 Mehlig 考虑的期望的最简单概括。它们也自然地出现在克拉科夫学派研究的具有复矩阵值的布朗运动的特征向量和特征值的联合演化问题中。我们发现这些期望具有决定性结构，其中相关核可以用复平面中的某些正交多项式来表示。此外，内核对所有 [公式：见正文] 都承认了一个相当容易处理的表达式。该结果可以相当直接地计算局部体积和边缘缩放限制中的重叠矩阵的条件期望，并证明这些期望在体积中的精确代数衰减和渐近分解。