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

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 8, page 1927-2036
- ISSN: 1435-9855

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topErdős, 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 -

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