The local semicircle law for a general class of random matrices
By L. Erdős, Antti Knowles, Horng-Tzer Yau, Jun Yin. 2012
230 citations recorded by the source corpus.
We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \mathbb{E} \left|h_{ij}\right|^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width $W\gg N^{1-\varepsilon_n}$ with some $\varepsilon_n>0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/aa85713d8941b8aa5beb68cac82cf9265a1abbb6
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Anisotropic local laws for random matrices (2014, background): “1007/s00220-012-1527-7 [14] Erdős, L; Knowles, A; Yau, H-T; Yin, J, The local semicircle law for a general class of random matrices, Electron.”
- Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs (2017, result): “near statistics, after normalizing by p1=2, converge to a Gaussian random variable. A three-step dynamical approach to the local statistics of random matrices has been developed in a series of papers [2,8,18,22{28,41,42]. This strategy is as follows. The work of H.-T. Y. is partially supported by NS”
- Extremal eigenvalues and eigenvectors of deformed Wigner matrices (2013, methodology): “Following [17], we pick a term Y ≡ Y (a,b1, .”
- Edge universality for deformed Wigner matrices (2014, background): “The following lemma, taken from [15], provides useful large deviation estimates.”
- A Dynamical Approach to Random Matrix Theory (2017, background): “4, we also remark that there is a considerable room in this argument even if optimal rigidity is not available (this is the case for random band matrices, see [55]).”
- Spiked separable covariance matrices and principal components (2019, methodology): “We will use the following notion of stochastic domination, which was first introduced in [19] and subsequently used in many works on random matrix theory, such as [7, 8, 9, 20, 21, 30].”
- Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, methodology): “Optimal local laws in the bulk and at the edges of the spectrum were obtained for Wigner matrices in [18, 17, 15].”
- Lectures on the local semicircle law for Wigner matrices (2016, unknown): “04 05 5v 3 [ m at h. PR ] 1 8 O ct 2 01 6 Contents”