2024/09/04 by Paul Bourgade, Bourgade, Paul, Giorgio Cipolloni +3 · 1 citation
Physics and Astronomy · Chemistry · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Molecular spectroscopy and chirality
paper · pdf · doi:10.48550/arxiv.2409.02902
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 Virág 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.