Fluctuations for non-Hermitian dynamics

Paul Bourgade; Giorgio Cipolloni; Jiaoyang Huang

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arXiv · 2409.02902

Fluctuations for non-Hermitian dynamics

Paul Bourgade·Giorgio Cipolloni·Jiaoyang Huang

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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 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.

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Paul Bourgade, Giorgio Cipolloni, Jiaoyang Huang. 2026-03-02. Fluctuations for non-Hermitian dynamics. https://arxiv.org/abs/2409.02902

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