Universality of local spectral statistics of random matrices
By L. Erdős, Horng-Tzer Yau. 2011
144 citations recorded by the source corpus.
The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given by a log-gas with inverse temperature $\beta = 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles. For $\beta \not \in \{1, 2, 4\}$, there is no matrix model behind this model, but the statistical physics interpretation of the log-gas is still valid for all $\beta > 0$. The universality conjecture for invariant ensembles asserts that the local eigenvalue statistics are independent of $V$. In this article, we review our recent solution to the universality conjecture for both invariant and non-invariant ensembles. We will also demonstrate that the local ergodicity of the Dyson Brownian motion is the intrinsic mechanism behind the universality. Furthermore, we review the solution of Dyson's conjecture on the local relaxation time of the Dyson Brownian motion. Related questions such as delocalization of eigenvectors and local version of Wigner's semicircle law will also be discussed.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/5331423a0f5b6bfa83caa9b7d7a162f39d43ec19
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In Discrete Mathematics and Combinatorics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Systems of Brownian particles with asymmetric collisions (2012, background): “3 in [1]) have played a central rôle in the understanding of the universality properties of large Hermitian random matrices with independent entries (Wigner matrices, see the survey [9] and the references there).”
- Random matrices: tail bounds for gaps between eigenvalues (2015, unknown): “of Wigner matrices; see [30, 10, 31] for surveys.”
- A necessary and sufficient condition for edge universality of Wigner matrices (2012, unknown): “See the reviews [22, 23] for further discussion.”
- No-gaps delocalization for general random matrices (2015, background): “Guided by the ubiquitous universality phenomenon in random matrix theory (see [33, 35, 18, 10]), we can anticipate that the eigenvectors behave in a similar way for a much broader class of random matrices.”
- The local semicircle law for a general class of random matrices (2012, background): “2 of [13]) shows, the local statistics of the eigenvalue gaps of Ht and GUE/GOE coincide if t > Nτ = NQ, i.”
- Delocalization of eigenvectors of random matrices with independent entries (2013, background): “that general symmetric and Hermitian random matrices H with independent entries are completely delocalized ([10, 11, 12, 14, 29], see also surveys [13, 4]).”
- Black holes and random matrices (2016, background): “Insights have emerged [72, 73, 74, 58] about the universality of dense random matrix behavior in the fine grained eigenvalue statistics of various types of sparse matrices.”
- Lectures on integrable probability (2012, background): “lity delivers universal behavior for the whole universality class of similar models. Moreover, in the rare occasions when the universality can be proved (e.g. in random matrices, see recent reviews 9 [Erdos-Yau-12], [Tao-Vu-12] and references therein, or in (1 + 1)d polymers in the socalled intermed”
- A short survey of Stein's method (2014, methodology): “Sophisticated forms of this idea have been used by Erdős, Yau and coauthors in their remarkable series of papers [42, 43, 44, 45] proving universality of random matrix eigenvalue distributions, running parallel to the papers of Tao and Vu, who used the Lindeberg approach.”
- Random matrix theory in statistics: A review (2014, methodology): “Significant new developments on the universality phenomena have been made by Erdős et al. (2010, 2009, 2012), Erdős (2010), Erdős and Yau (2012), and Tao and Vu (2010a,b, 2011, 2012a,b) who used analytical techniques to study the question of both bulk and edge universality and managed to remove…”
- Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane (2015, background): “Second, the recent proofs of the Wigner-Dyson-Mehta conjecture on the universality of local eigenvalue statistics heavily rely on understanding the spectral density on very small scales (see [18] for a complete history).”