论文标题
特征值在变形的Ginibre合奏中的相变
Phase transition of eigenvalues in deformed Ginibre ensembles
论文作者
论文摘要
将大小$ n $的随机矩阵作为确定性矩阵$ x_0 $具有有限等级的复杂ginibre集合的添加变形，独立于$ n $。当$ x_0 $的某些特征值与单位磁盘分开时，离群特征值可能在同一位置渐近，并且它们的波动表现出令人惊讶的现象，这些现象高度依赖于约旦规范形式的$ x_0 $。这些发现很大程度上是由于Benaych-Georges和Rochet \ Cite {Br}，Bordenave和Capitaine \ Cite \ Cite {BC16}，以及Tao \ cite {TA13}。当$ x_0 $的所有特征值位于设备磁盘内时，我们证明，光谱边缘处的本地特征值统计数据形成了一类新的确定点过程，为此，相关内核是根据重复的ERFC积分来表征的。因此，这在随机矩阵理论中完成了BBP相变的非热类似物。变形的真实四基因果乐团的类似结果。
Consider a random matrix of size $N$ as an additive deformation of the complex Ginibre ensemble under a deterministic matrix $X_0$ with a finite rank, independent of $N$. When some eigenvalues of $X_0$ separate from the unit disk, outlier eigenvalues may appear asymptotically in the same locations, and their fluctuations exhibit surprising phenomena that highly depend on the Jordan canonical form of $X_0$. These findings are largely due to Benaych-Georges and Rochet \cite{BR}, Bordenave and Capitaine \cite{BC16}, and Tao \cite{Ta13}. When all eigenvalues of $X_0$ lie inside the unit disk, we prove that local eigenvalue statistics at the spectral edge form a new class of determinantal point processes, for which correlation kernels are characterized in terms of the repeated erfc integrals. This thus completes a non-Hermitian analogue of the BBP phase transition in Random Matrix Theory. Similar results hold for the deformed real quaternion Ginibre ensemble.
