@article{429,
  abstract     = {We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.},
  author       = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
  issn         = {14322064},
  journal      = {Probability Theory and Related Fields},
  number       = {1-2},
  pages        = {293–373},
  publisher    = {Springer},
  title        = {{Stability of the matrix Dyson equation and random matrices with correlations}},
  doi          = {10.1007/s00440-018-0835-z},
  volume       = {173},
  year         = {2019},
}

@article{721,
  abstract     = {Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.},
  author       = {Ajanki, Oskari H and Krüger, Torben H and Erdös, László},
  issn         = {00103640},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {9},
  pages        = {1672 -- 1705},
  publisher    = {Wiley-Blackwell},
  title        = {{Singularities of solutions to quadratic vector equations on the complex upper half plane}},
  doi          = {10.1002/cpa.21639},
  volume       = {70},
  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{1489,
  abstract     = {We prove optimal local law, bulk universality and non-trivial decay for the off-diagonal elements of the resolvent for a class of translation invariant Gaussian random matrix ensembles with correlated entries. },
  author       = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
  journal      = {Journal of Statistical Physics},
  number       = {2},
  pages        = {280 -- 302},
  publisher    = {Springer},
  title        = {{Local spectral statistics of Gaussian matrices with correlated entries}},
  doi          = {10.1007/s10955-016-1479-y},
  volume       = {163},
  year         = {2016},
}

@article{2179,
  abstract     = {We extend the proof of the local semicircle law for generalized Wigner matrices given in MR3068390 to the case when the matrix of variances has an eigenvalue -1. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices X*X, where the variances of the entries of X may vary.},
  author       = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
  journal      = {Electronic Communications in Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Local semicircle law with imprimitive variance matrix}},
  doi          = {10.1214/ECP.v19-3121},
  volume       = {19},
  year         = {2014},
}