# 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

, Volume 355 (3) – Aug 9, 2017
52 pages

/lp/springer_journal/convergence-of-local-statistics-of-dyson-brownian-motion-vKg0FNc0jD
Publisher
Springer Berlin Heidelberg
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations