Fluctuations for non-Hermitian dynamics

TL;DR

This paper demonstrates that the log-determinant of large non-Hermitian matrices under Brownian evolution converges to a 2+1D Gaussian field with logarithmic correlations, extending previous results to out-of-equilibrium dynamics.

Contribution

It establishes the convergence of the log-determinant to a Gaussian field in a non-Hermitian setting, revealing non-Markovian properties and long-range eigenvector correlations.

Findings

Log-determinant converges to a 2+1D Gaussian field

The limiting field exhibits logarithmic correlations

Eigenvector overlaps have polynomial decay in space-time

Abstract

We prove that under the Brownian evolution on large non-Hermitian matrices the log-determinant converges in distribution to a 2+1 dimensional Gaussian field in the Edwards-Wilkinson regularity class, namely it is logarithmically correlated for the parabolic distance. This dynamically extends a seminal result by Rider and Vir\'ag about convergence to the Gaussian free field. The convergence holds out of equilibrium for centered, i.i.d. matrix entries as an initial condition. A remarkable aspect of the limiting field is its non-Markovianity, due to long range correlations of the eigenvector overlaps, for which we identify the exact space-time polynomial decay. In the proof, we obtain a quantitative, optimal relaxation at the hard edge, for a broad extension of the Dyson Brownian motion, with a driving noise arbitrarily correlated in space.