Eikonal formulation of large dynamical random matrix models

The standard approach to dynamical random matrix models relies on the description of trajectories of eigenvalues. Using the analogy from optics, based on the duality between the Fermat principle (rays) and the Huygens principle (wavefronts), we formulate the Hamilton-Jacobi dynamics for large random matrix models. The resulting equations describe a broad class of random matrix models in a unified way, including normal (Hermitian or unitary) as well as strictly non-normal dynamics. This formalism applied to Brownian bridge dynamics allows one to calculate the asymptotics of the Harish-Chandra-Itzykson-Zuber integrals.

Figure 1

The construction of the duality between wavefronts (black solid lines) and rays (dotted gray arrows) in (a) optics and (b) dynamical random matrices is highly analogous. Optical wavefronts in real space trace light-ray propagation, while matrix wavefronts $\mathrm{\Phi }(z,t)=\text{const}$ trace complex plane propagation of the characteristics. In the latter, the insets show a highly anisotropic wavefront and ray evolution when projected along real and imaginary axes. The presented random matrix dynamics is given by the Gaussian diffusion with zero initial condition giving rise to GUE. Details are given in Appendix pp1.

Figure 2

Construction of duality between wavefronts (black solid lines) and rays (dotted gray arrows) in non-Hermitian dynamical random matrices. Propagation takes place in restricted quaternionic space spanned by modules $\left|z\right|$ and $\left|w\right|=r$. Wavefronts are defined by condition $\mathrm{\Phi }\left(r\right(t),t)=\text{const}$, with the potential given by Eq. (B7) and characteristics $z\left(t\right)=z_0,r\left(t\right)=r_0-r_0/\left(\right|z_0{|}^2+r_0^2)t$ by Eq. (B5). The $z$ variable is trivially added despite the absence of interesting dynamics. Similarly to Fig. 1, propagation is likewise anisotropic. The presented evolution happens for the Ginibre dynamics