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Fluctuations for non-Hermitian dynamics

Paul Bourgade, Giorgio Cipolloni, Jiaoyang Huang

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.

Fluctuations for non-Hermitian dynamics

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.
Paper Structure (34 sections, 28 theorems, 279 equations, 1 figure)

This paper contains 34 sections, 28 theorems, 279 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Random matrices and logarithmically correlated fields, dimensions one and two.
  3. Result.
  4. Related literature.
  5. Acknowledgments.
  6. Main Results.
  7. Multi-time central limit theorems for eigenvalues.
  8. Eigenvector overlaps.
  9. Proof ideas.
  10. Outline.
  11. Notations.
  12. Time correlations: Proof of Theorems \ref{['theo:mainres']}--\ref{['theo:mainresmeso']}
  13. Decomposition in three terms.
  14. Submicroscopic scale.
  15. Microscopic scale.
  16. ...and 19 more sections

Key Result

Theorem 1

Consider (eq:OU) at equilibrium and let $f,g$ be smooth functions supported in the open unit disk $\mathbb{D}$. Then for any fixed $s,t$, as $N\to\infty$ the linear statistics ${\rm Tr} f(G_s)-N\int \frac{f}{\pi}$ and ${\rm Tr} g(G_t)-N\int \frac{g}{\pi}$ converge jointly to Gaussian random variable

Figures (1)

  • Figure 1: $\Lambda_t$ as defined in \ref{['mtlambdat']} is an open subset of the upper half plane $\mathbb H$. $\Phi(z)$ is a holomorphic map from $\Lambda_t$ to $\mathbb H$.

Theorems & Definitions (61)

  • Theorem
  • Theorem 2.2: Macroscopic CLT
  • Theorem 2.3: Mesoscopic CLT
  • Remark 2.4: Mixing and distance to the edge
  • Remark 2.5: Averaging by stereographic projection
  • Remark 2.6: Existence of Gaussian fields
  • Remark 2.7: Restriction to the cylinder
  • Remark 2.8: Uniformity in the parameters
  • Remark 2.9: Coupling at equilibrium
  • Proposition 2.10
  • ...and 51 more