Full Dysonian dynamics of the complex Ginibre ensemble

TL;DR

This paper derives stochastic equations for eigenvalues and eigenvectors of the dynamical complex Ginibre ensemble, providing solutions and new formulas for correlation functions, with detailed analysis of initial conditions and limiting behaviors.

Contribution

It introduces a comprehensive dynamical framework for the complex Ginibre ensemble, including stochastic equations and explicit solutions for correlation functions.

Findings

Derived stochastic equations for eigenvalues and eigenvectors.

Solved the Smoluchowski-Fokker-Planck equation for any initial matrix.

Obtained new formulas for eigenvector correlation functions and analyzed their limits.

Abstract

We find stochastic equations governing eigenvalues and eigenvectors of a dynamical complex Ginibre ensemble reaffirming the intertwined role played between both sets of matrix degrees of freedom. We solve the accompanying Smoluchowski-Fokker-Planck equation valid for any initial matrix. We derive evolution equations for the averaged extended characteristic polynomial and for a class of $k$-point eigenvalue correlation functions. From the latter we obtain a novel formula for the eigenvector correlation function which we inspect for Ginibre and spiric initial conditions and obtain macro- and microscopic limiting laws.