Unveiling the significance of eigenvectors in diffusing non-Hermitian matrices by identifying the underlying Burgers dynamics

Abstract

Following our recent letter [1], we study in detail an entry-wise diffusion of non-hermitian complex matrices. We obtain an exact partial differential equation (valid for any matrix size N and arbitrary initial conditions) for evolution of the averaged extended characteristic polynomial. The logarithm of this polynomial has an interpretation of a potential which generates a Burgers dynamics in quaternionic space. The dynamics of the ensemble in the large N limit is completely determined by the coevolution of the spectral density and a certain eigenvector correlation function. This coevolution is best visible in an electrostatic potential of a quaternionic argument built of two complex variables, the first of which governs standard spectral properties while the second unravels the hidden dynamics of eigenvector correlation function. We obtain general formulas for the spectral density and the eigenvector correlation function for large N and for any initial conditions. We exemplify our studies by solving three examples, and we verify the analytic form of our solutions with numerical simulations.

Ornstein–Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links

“…The presented results borrow to a large extent from the conclusions obtained in the series of papers of the present authors [12,18,29,30], but also include new solutions. First, we adapted the turbulent scenario to the Ornstein-Uhlenbeck process for GUE.…”

On the determinantal structure of conditional overlaps for the complex Ginibre ensemble

“…It is not uncommon that an approach that uses orthogonal polynomials for finite-N is more difficult when it comes to deriving global correlation functions, both for eigenvalues and eigenvectors. Here, the methods of Green's functions [10,11] and Feynman diagrams [13] advocated by the Krakow group have proven much more useful. We refer to [13] for a comprehensive list of global bulk correlations of the off-diagonal overlap in various ensembles, see also the references therein.…”

“…Below we give an incomplete list of results for different ensembles of random matrices, with the complex Ginibre ensembles introduced in [9] being most studied. Using a combination of Green's functions and diffusion equations, it was noticed early on that the Dysonian dynamics in this ensemble couples the complex eigenvalues and their eigenvectors in a non-trivial way [10,11]. These techniques were further developed including Feynman diagrams [12,13], free probability [14] or stochastic differential equations [15] and applied to different ensembles including products of elliptic Ginibre matrices [12].…”

“…viewed as functions in all their independent 2k variables are entire.3. Results at Finite-NThe first observation made in[28] is a simple operation relating D (N,k)11 and D (N,k) 12…”

Full Dysonian dynamics of the complex Ginibre ensemble

“…This approach was more recently adapted to derive and solve partial differential equations describing the dynamics of the resolvent (or Greens function) and the averaged characteristic polynomial for diffusing Hermitian [15,16] and Wishart matrices [17,18]. More importantly for this paper, this was extended to the study of complex non-Hermitian matrices [19,20], where the intimate interplay of the introduced correlation function (1) and the spectral resolvent was uncovered in the form of a coupling of associated nonlinear partial differential equations. These were derived by inspecting dynamic properties of a novel, extended form of an associated averaged characteristic polynomial, which turned out to satisfy a very simple diffusion equation in an auxiliary spatial-like dimension.…”