Fluctuations for non-Hermitian dynamics

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.

💡 Research Summary

The paper studies the stochastic evolution of large non‑Hermitian matrices under a complex Ornstein‑Uhlenbeck (Brownian) flow. Starting from an N×N matrix with centered i.i.d. entries of unit variance, the dynamics are given by dX_t = dB_t/√N – (1/2)X_t dt, where B_t has independent complex Brownian entries. This flow has the Ginibre ensemble as its unique invariant measure. The authors focus on the logarithm of the characteristic polynomial, equivalently the linear statistics L_N(f,t)=∑_{i=1}^N f(σ_i(t))−E∑ f(σ_i(t)), where σ_i(t) are the eigenvalues of X_t and f is a test function.

The main achievement is a full space‑time central limit theorem for these linear statistics, both on macroscopic and mesoscopic scales. For any smooth test functions f,g supported in a fixed disk containing the spectrum, and for any fixed times s≤t, the pair (L_N(f,t), L_N(g,s)) converges jointly to a centered Gaussian vector (L(f,t), L(g,s)). The covariance is expressed through a kernel \