Gap universality of generalized Wigner and $\beta$-ensembles

Erdős, László, and Yau, Horng-Tzer. "Gap universality of generalized Wigner and $\beta $-ensembles." Journal of the European Mathematical Society 017.8 (2015): 1927-2036. <http://eudml.org/doc/277176>.

@article{Erdős2015,
abstract = {We consider generalized Wigner ensembles and general $\beta $-ensembles with analytic potentials for any $\beta \ge 1 $. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian $\beta $-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any potential $C^4(\mathbb \{R\})$.},
author = {Erdős, László, Yau, Horng-Tzer},
journal = {Journal of the European Mathematical Society},
keywords = {$\beta $-ensembles; Wigner-Dyson-Gaudin-Mehta universality; gap distribution; log-gas; Wigner ensembles; -ensembles; Wigner-Dyson-Gaudin-Mehta universality; gap distributions; Dyson Brownian motion; De Giorgi-Nash-Moser method; Gaussian matrices; log-gas Hamiltonians; random matrices},
language = {eng},
number = {8},
pages = {1927-2036},
publisher = {European Mathematical Society Publishing House},
title = {Gap universality of generalized Wigner and $\beta $-ensembles},
url = {http://eudml.org/doc/277176},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Erdős, László
AU - Yau, Horng-Tzer
TI - Gap universality of generalized Wigner and $\beta $-ensembles
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 8
SP - 1927
EP - 2036
AB - We consider generalized Wigner ensembles and general $\beta $-ensembles with analytic potentials for any $\beta \ge 1 $. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian $\beta $-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any potential $C^4(\mathbb {R})$.
LA - eng
KW - $\beta $-ensembles; Wigner-Dyson-Gaudin-Mehta universality; gap distribution; log-gas; Wigner ensembles; -ensembles; Wigner-Dyson-Gaudin-Mehta universality; gap distributions; Dyson Brownian motion; De Giorgi-Nash-Moser method; Gaussian matrices; log-gas Hamiltonians; random matrices
UR - http://eudml.org/doc/277176
ER -