Convergence of Local Statistics of Dyson Brownian Motion

Convergence of Local Statistics of Dyson Brownian Motion We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times t = o(1) with deterministic initial data V. Our main result states that if the density of states of V is bounded both above and away from 0 down to scales $${\ell \ll t}$$ ℓ ≪ t in a small interval of size $${G \gg t}$$ G ≫ t around an energy $${E_0}$$ E 0 , then the local statistics coincide with the GOE/GUE near the energy $${E_0}$$ E 0 after time t. Our methods are partly based on the idea of coupling two Dyson Brownian motions from Bourgade et al. (Commun Pure Appl Math, 2016), the parabolic regularity result of Erdős and Yau (J Eur Math Soc 17(8):1927–2036, 2015), and the eigenvalue rigidity results of Lee and Schnelli (J Math Phys 54(10):103504, 2013). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

Convergence of Local Statistics of Dyson Brownian Motion

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-017-2955-1
Publisher site
See Article on Publisher Site

Abstract

We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times t = o(1) with deterministic initial data V. Our main result states that if the density of states of V is bounded both above and away from 0 down to scales $${\ell \ll t}$$ ℓ ≪ t in a small interval of size $${G \gg t}$$ G ≫ t around an energy $${E_0}$$ E 0 , then the local statistics coincide with the GOE/GUE near the energy $${E_0}$$ E 0 after time t. Our methods are partly based on the idea of coupling two Dyson Brownian motions from Bourgade et al. (Commun Pure Appl Math, 2016), the parabolic regularity result of Erdős and Yau (J Eur Math Soc 17(8):1927–2036, 2015), and the eigenvalue rigidity results of Lee and Schnelli (J Math Phys 54(10):103504, 2013).

Journal

Communications in Mathematical PhysicsSpringer Journals

Published: Aug 9, 2017

References

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