Fluctuations for non-Hermitian dynamics
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914361994379264 |
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| author | Bourgade, Paul Cipolloni, Giorgio Huang, Jiaoyang |
| author_facet | Bourgade, Paul Cipolloni, Giorgio Huang, Jiaoyang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fluctuations for non-Hermitian dynamics Bourgade, Paul Cipolloni, Giorgio Huang, Jiaoyang Probability 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. |
| title | Fluctuations for non-Hermitian dynamics |
| topic | Probability |
| url | https://arxiv.org/abs/2409.02902 |