Phase transition phenomenon in deformed GinUE
A talk in the Seminar Zufallsmatrizen series by
Lu Zhang
| Abstract: | 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 $N$. We observe a phase transition for the extreme eigenvalues(in the sense of modulus) in deformed GinUE, both in eigenvalue statistics(via 1-point correlation function) and eigenvector statistics(via the mean self overlap function). There are three regimes: (1) When all eigenvalues of $X_0$ lie in the open disk $D(0,\sqrt{\tau})$, local statistics are still governed by the GinUE statistics; (2) When some eigenvalues of $X_0$ are on the circle and all others lie inside it, both eigenvalue and eigenvector statistics can be characterized by the iterative erfc integrals; (3) When some eigenvalues of $X_0$ go outside the circle, outlier eigenvalues occur, and the eigenvalue and eigenvector statistics are characterized by new kinds of functions which are different from the edge statistics. The contents are mainly based on "Phase transition of eigenvalues in deformed Ginibre ensembles. arXiv:2204.13171v2", joint work with Dang-Zheng Liu and "Mean eigenvector self-overlap in deformed complex Ginibre ensemble. arXiv:2407.09163v2". Within the CRC this talk is associated to the project(s): C6 |