@article{550,
abstract = {For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX* is well-approximated by a deterministic measure on ℝ. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities.},
author = {Alt, Johannes},
issn = {1083589X},
journal = {Electronic Communications in Probability},
publisher = {Institute of Mathematical Statistics},
title = {{Singularities of the density of states of random Gram matrices}},
doi = {10.1214/17-ECP97},
volume = {22},
year = {2017},
}
@book{567,
abstract = {This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality.
},
author = {Erdös, László and Yau, Horng},
isbn = {9781470436483},
pages = {226},
publisher = {American Mathematical Society},
title = {{A dynamical approach to random matrix theory}},
volume = {28},
year = {2017},
}
@article{615,
abstract = {We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law.},
author = {Erdös, László and Schnelli, Kevin},
issn = {02460203},
journal = {Annales de l'institut Henri Poincare (B) Probability and Statistics},
number = {4},
pages = {1606 -- 1656},
publisher = {Institute of Mathematical Statistics},
title = {{Universality for random matrix flows with time dependent density}},
doi = {10.1214/16-AIHP765},
volume = {53},
year = {2017},
}
@article{1337,
abstract = {We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.},
author = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
issn = {01788051},
journal = {Probability Theory and Related Fields},
number = {3-4},
pages = {667 -- 727},
publisher = {Springer},
title = {{Universality for general Wigner-type matrices}},
doi = {10.1007/s00440-016-0740-2},
volume = {169},
year = {2017},
}
@article{1528,
abstract = {We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.},
author = {Bao, Zhigang and Erdös, László},
issn = {01788051},
journal = {Probability Theory and Related Fields},
number = {3-4},
pages = {673 -- 776},
publisher = {Springer},
title = {{Delocalization for a class of random block band matrices}},
doi = {10.1007/s00440-015-0692-y},
volume = {167},
year = {2017},
}