Dysonian Dynamics of the Ginibre Ensemble

Abstract

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a non-trivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the non-analytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics, that we observe for the Ginibre ensemble, is a general feature of non-Hermitian random matrix models and is relevant to related physical applications. On the other hand, the original Dyson's idea of Brownian walk of real eigenvalues [11] interacting with a twodimensional Coulombic force still leads to novel insights. Examples include the study of determinantal processes [12][13][14], Loewner diffusion [15] or the fluctuations of nonintersecting interfaces in thermal equilibrium [16]. The concept of stochastic evolution of matrices has been recently exploited by several authors [17][18][19][20]. In particular, it was shown that the derivatives of the logarithms of characteristic determinants of diffusing GUE (Gaussian Unitary Ensemble), LUE (Laguerre Unitary Ensemble or Wishart Ensemble) and CUE (Circular Unitary Ensemble) obey Burgers-like nonlinear equations, where the role of viscosity is played by the inverse of the matrix size. For infinite dimensions of the matrix, these equations correspond to the inviscid regime and describe an evolution of the associated resolvents. Due to nonlinearity, they develop singularities (shock waves), whose positions correspond to the endpoints of the spectra. For matrices of finite size, the expansion around the shock wave solution of the initial viscid Burgers equation leads to a universal scaling of characteristic polynomials (and of the inverse characteristic polynomials as well), resulting in well known universal oscillatory behavior of the Airy, Bessel or Pearcey type. This approach has prompted, in particular, new perception of weak/strong coupling transition in multicolor Yang-Mills theory [21,22] and of the spontaneous breakdown of chiral symmetry in Euclidean QCD [23].In this letter, we unveil the intertwined evolution of eigenvalues and eigenvectors of stochastically evolving non-Hermitian matrices. To this end, we apply Dyson's idea to study diffusing Gaussian matrices for the case of the Ginibre Ensemble (GE). The central object of the paper is a generalized averaged characteristic polynomial. Its logarithmic derivatives, which contain the information about both the eigenvalues and eigenvectors of the evolving matrix, fulfill a sys...

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