@article{1010,
  abstract     = {We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. },
  author       = {Alt, Johannes and Erdös, László and Krüger, Torben H},
  issn         = {10836489},
  journal      = {Electronic Journal of Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Local law for random Gram matrices}},
  doi          = {10.1214/17-EJP42},
  volume       = {22},
  year         = {2017},
}

@article{1023,
  abstract     = {We consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε.},
  author       = {Nemish, Yuriy},
  issn         = {10836489},
  journal      = {Electronic Journal of Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Local law for the product of independent non-Hermitian random matrices with independent entries}},
  doi          = {10.1214/17-EJP38},
  volume       = {22},
  year         = {2017},
}

@article{2225,
  abstract     = {We consider sample covariance matrices of the form X∗X, where X is an M×N matrix with independent random entries.  We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent (X∗X−z)−1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity ⟨v,(X∗X−z)−1w⟩−⟨v,w⟩m(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v,w∈CN. We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Iz≥N−1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
},
  author       = {Bloemendal, Alex and Erdös, László and Knowles, Antti and Yau, Horng and Yin, Jun},
  issn         = {10836489},
  journal      = {Electronic Journal of Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Isotropic local laws for sample covariance and generalized Wigner matrices}},
  doi          = {10.1214/EJP.v19-3054},
  volume       = {19},
  year         = {2014},
}