A necessary and sufficient condition for edge universality of Wigner matrices
By Ji Oon Lee, Jun Yin. 2012
139 citations recorded by the source corpus.
In this paper, we prove a necessary and sufficient condition for Tracy-Widom law of Wigner matrices. Consider $N \times N$ symmetric Wigner matrices $H$ with $H_{ij} = N^{-1/2} x_{ij}$, whose upper right entries $x_{ij}$ $(1\le i< j\le N)$ are $i.i.d.$ random variables with distribution $\mu$ and diagonal entries $x_{ii}$ $(1\le i\le N)$ are $i.i.d.$ random variables with distribution $\wt \mu$. The means of $\mu$ and $\wt \mu$ are zero, the variance of $\mu$ is 1, and the variance of $\wt \mu $ is finite. We prove that Tracy-Widom law holds if and only if $\lim_{s\to \infty}s^4\p(|x_{12}| \ge s)=0$. The same criterion holds for Hermitian Wigner matrices.
Subfields: Algebra and Number Theory, Management Information Systems, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/9e68911833aa2b30f4a25aee1a8231146b589870
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Citations in the corpus
- A Dynamical Approach to Random Matrix Theory (2017, methodology): “In alphabetical order, these coauthors were Paul Bourgade, Antti Knowles, Sandrine Péché, Jose Ramı́rez, Benjamin Schlein and Jun Yin.”
- Extremal eigenvalues and eigenvectors of deformed Wigner matrices (2013, background): “Recently, a necessary and sufficient condition for the edge universality of Wigner matrices was given in [33].”
- A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, methodology): “The authors would like to thank Jeremy Quastel and Jun Yin for fruitful discussions and valuable suggestions, which have significantly improved the paper.”
- Random matrix theory in statistics: A review (2014, background): “Lee and Yin (2012) established necessary and sufficient conditions for edge universality of a Wigner matrix.”
- Dyson Brownian motion for general $$\beta $$ and potential at the edge (2018, methodology): “Local laws for β-ensembles near the edge were considered in [5] with the discrete analogue in [26]; the Wigner matrices were considered in [32].”
- Edge universality for deformed Wigner matrices (2014, background): “Recently, a necessary and sufficient condition on the entries’ distribution for the edge universality of Wigner matrices was given in [27].”
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, methodology): “off-diagonal entries (up to symmetry) was established by Lee and Yin in [31], where the matrix entry is only required to have mean 0 and variance 1, and satisfies a tail condition slightly weaker than the existence of the 4th moment.”
- Edge Universality of Beta Ensembles (2013, background): “[19] L. Erdős, A. Knowles, H.-T. Yau, and J. Yin, Spectral statistics of Erdős-Rényi Graphs II: Eigenvalue spacing and the extreme eigenvalues, Comm. Math.”