Random matrices, log-gases and Holder regularity
arXiv:1407.5752
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.
arXiv commentProceedings of ICM 2014
References in corpus (16)
- Level-Spacing Distributions and the Airy Kernel
- Local semicircle law and complete delocalization for Wigner random matrices
- Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices
- Continuum limits of random matrices and the Brownian carousel
- Edge Universality of Beta Ensembles
- Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
- Eigenvector localization for random band matrices with power law band width
- Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
- Fluctuations of eigenvalues of matrix models and their applications
- Orthogonal and symplectic matrix models: universality and other properties
- Universality of the local regime for the block band matrices with a finite number of blocks
- On the edge universality of the local eigenvalue statistics of matrix models
- Fixed energy universality for generalized Wigner matrices
- Transport maps for Beta-matrix models and Universality
- On the second mixed moment of the characteristic polynomials of the 1D band matrices
- The asymptotic distribution of a single eigenvalue gap of a Wigner matrix