Abstract
We consider the non-hermitian matrix-valued process of Elliptic Ginibre Ensemble. This model includes Dyson's Brownian motion model and the time evolution model of Ginibre ensemble by using hermiticity parameter. We show the complex eigenvalue processes satisfy the stochastic differential equations which are very similar to Dyson's model and give an explicit form of overlap correlations. As a corollary, in the case of 2-by-2 matrix, we also mention the relation between the diagonal overlap, which is the speed of eigenvalues, and the distance of the two eigenvalues.
Highlights
The study of eigenvalue processes was started by Dyson [11]
The main result of this paper is to give the stochastic differential equations (2.5), with parameter −1 ≤ τ ≤ 1, that the eigenvalue processes of Elliptic Ginibre ensemble (EGE) satisfy
In the main theorem (Theorem 2.1), we show that for −1 ≤ τ ≤ 1, the eigenvalue processes of EGE satisfy the stochastic differential equations which have the drift of Dyson’s model except for τ = 0 and show the explicit form of their timedepending overlaps described by given Brownian motions
Summary
The study of eigenvalue processes was started by Dyson [11]. There are remarkable results for Ginibre ensemble This model is non-symmetric matrix-valued process whose entries are given by independent complex Brownian motions. On the basis of the above results and observations, we consider the non-normal matrix valued process of EGE whose entries are given by independent Brownian motions This model naturally gives an interpolation between Dyson’s model and the time evolution of Ginibre ensemble by using hermiticity parameter τ in the same way as the statistic case. In the main theorem (Theorem 2.1), we show that for −1 ≤ τ ≤ 1, the eigenvalue processes of EGE satisfy the stochastic differential equations which have the drift of Dyson’s model except for τ = 0 and show the explicit form of their timedepending overlaps described by given Brownian motions. We put together some properties of characteristic polynomials, eigenvalues and determinants in Appendix
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