# Random matrices, log-gases and Hölder regularity

Erdös L. 2014. Random matrices, log-gases and Hölder regularity. Proceedings of the International Congress of Mathematicians. ICM: International Congress of Mathematicians vol. 3, 214–236.

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http://arxiv.org/abs/1407.5752
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Abstract

The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner’s original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices.

Publishing Year

Date Published

2014-08-01

Proceedings Title

Proceedings of the International Congress of Mathematicians

Acknowledgement

The author is partially supported by SFB-TR 12 Grant of the German Research Council.

Volume

3

Page

214 - 236

Conference

ICM: International Congress of Mathematicians

Conference Location

Seoul, Korea

Conference Date

2014-08-13 – 2014-08-21

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### Cite this

Erdös L. Random matrices, log-gases and Hölder regularity. In:

*Proceedings of the International Congress of Mathematicians*. Vol 3. International Congress of Mathematicians; 2014:214-236.Erdös, L. (2014). Random matrices, log-gases and Hölder regularity. In

*Proceedings of the International Congress of Mathematicians*(Vol. 3, pp. 214–236). Seoul, Korea: International Congress of Mathematicians.Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” In

*Proceedings of the International Congress of Mathematicians*, 3:214–36. International Congress of Mathematicians, 2014.L. Erdös, “Random matrices, log-gases and Hölder regularity,” in

*Proceedings of the International Congress of Mathematicians*, Seoul, Korea, 2014, vol. 3, pp. 214–236.Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.”

*Proceedings of the International Congress of Mathematicians*, vol. 3, International Congress of Mathematicians, 2014, pp. 214–36.**All files available under the following license(s):**

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