Program

Abstract: The complex Ginibre matrix plays a fundamental role in non-Hermitian random matrix theory. In this talk, I will discuss the complex Ginibre matrix conditioned to have a deterministic eigenvalue at a specific point. This model is known as the induced ensemble and is also referred to as an insertion of a point charge from a statistical physics perspective. I will present recent developments on the eigenvalue correlation functions and the expansion of the partition functions for this model. The former relates to local universality questions, while the latter relates to the moments of the characteristic polynomial of the Ginibre matrix, which also finds applications in the large deviations of the Laguerre unitary ensemble. 

Abstract: This talk introduces a moment method to study mean-field limit of three stochastic processes related to three classical beta ensembles on the real line: Gaussian beta ensembles, Laguerre beta ensembles and Jacobi beta ensembles. The moment method works well even in case the parameter beta varies with the system size. Results at the process level imply recursive relations for moments of the limiting measures in all three cases. It is based on joint works with F. Nakano (Tohoku University) and H.D. Trinh (University of Science, Vietnam National University Hanoi).

Abstract: The non-Hermitian matrix-valued Brownian motion (BM) is the stochastic process of a random matrix whose entries are given by independent complex BMs. We consider a one-parameter extension of this process, which can be regarded as a dynamical version of Girko's elliptic ensemble interpolating the GUE and the Ginibre ensemble of random matrices. Notice that the eigenvalue process of GUE is known as Dyson's BM. For each process in this family, the bi-orthogonality relation is imposed between the left and the right eigenvector processes, which allows for the scale-transformation invariance of the system. There each eigenvalue process is coupled with the eigenvector-overlap process which is a Hermitian matrix-valued process with entries given by products of overlaps of the left and the right eigenvectors. Time-evolutionary point processes (PPs) of eigenvalues and the marked PPs weighted by the elements of eigenvector-overlap processes are introduced. We study the stochastic equations for the pairings of these PPs and proper test functions. This talk is based on the joint work with Syota Esaki (Oita), Satoshi Yabuoku (Kitakyushu), and Jacek Małecki (Wrocław). 

Abstract: In this talk, we consider stochastic processes associated with eigenvalues and eigenvectors of the non-Hermitian matrix-valued Brownian motion (nHBM). Each eigenvalue process is coupled with the eigenvector-overlap process, which is a Hermitian matrix-valued process with entries given by products of overlaps of the right and left eigenvectors. To analyze the dynamics of the eigenvalues, we think it is important to treat time-dependent point processes of eigenvalues and their variations weighted by the diagonal elements of the eigenvector-overlap processes. In this talk, we introduce tensor-valued processes associated with left and right eigenvectors and related time-dependent point processes. In addition, we discuss a relation to the time-dependent point process related to eigenvector-overlap processes. The present talk is based on the joint work with Makoto Katori (Chuo University), Jacek Małecki (Wrocław University of Science and Technology) and Satoshi Yabuoku (National Institute of Technology, Kitakyushu College).